From 402f1734934642517635aea923983ec38af812c5 Mon Sep 17 00:00:00 2001 From: felixwellen Date: Fri, 17 Nov 2023 18:49:39 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20agda/cub?= =?UTF-8?q?ical@e9b8b40fec2011ac9e74c479aaf886b370548550=20=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Cubical.Algebra.AbGroup.Base.html | 8 +- Cubical.Algebra.AbGroup.TensorProduct.html | 14 +- Cubical.Algebra.Algebra.Base.html | 2 +- Cubical.Algebra.Algebra.Properties.html | 6 +- Cubical.Algebra.Algebra.Subalgebra.html | 2 +- ...al.Algebra.CommAlgebra.FPAlgebra.Base.html | 6 +- ...ommAlgebra.FreeCommAlgebra.Properties.html | 6 +- ...Algebra.CommAlgebra.Instances.Initial.html | 4 +- ...al.Algebra.CommAlgebra.Instances.Unit.html | 4 +- Cubical.Algebra.CommAlgebra.Localisation.html | 6 +- ...gebra.CommAlgebra.LocalisationAlgebra.html | 4 +- Cubical.Algebra.CommAlgebra.Properties.html | 8 +- 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...al.Categories.Functor.ComposeProperty.html | 190 +++--- Cubical.Categories.Functor.Properties.html | 160 ++--- Cubical.Categories.Functors.Constant.html | 12 +- Cubical.Categories.Functors.HomFunctor.html | 12 +- Cubical.Categories.Instances.Categories.html | 6 +- ...cal.Categories.Instances.CommAlgebras.html | 210 +++--- Cubical.Categories.Instances.CommRings.html | 68 +- Cubical.Categories.Instances.Discrete.html | 12 +- ...l.Categories.Instances.EilenbergMoore.html | 130 ++-- ....Categories.Instances.FunctorAlgebras.html | 64 +- ...al.Categories.Instances.Functors.Endo.html | 4 +- Cubical.Categories.Instances.Functors.html | 122 ++-- Cubical.Categories.Instances.Groups.html | 77 +++ Cubical.Categories.Instances.Sets.html | 38 +- Cubical.Categories.Instances.Terminal.html | 12 +- Cubical.Categories.Instances.ZFunctors.html | 220 +++---- Cubical.Categories.Isomorphism.html | 28 +- Cubical.Categories.Limits.Initial.html | 24 +- Cubical.Categories.Limits.Limits.html | 148 ++--- Cubical.Categories.Limits.Pullback.html | 42 +- Cubical.Categories.Limits.RightKan.html | 202 +++--- Cubical.Categories.Limits.Terminal.html | 12 +- Cubical.Categories.Monad.Base.html | 46 +- Cubical.Categories.Monoidal.Base.html | 22 +- ...cal.Categories.Monoidal.Strict.Monoid.html | 6 +- ...Categories.NaturalTransformation.Base.html | 136 ++-- ...ries.NaturalTransformation.Properties.html | 323 +++++----- Cubical.Categories.Presheaf.Base.html | 16 +- Cubical.Categories.Presheaf.KanExtension.html | 190 +++--- Cubical.Categories.Presheaf.Morphism.html | 26 +- ...ategories.Presheaf.NonPresheaf.Cofree.html | 24 +- ...ategories.Presheaf.NonPresheaf.Forget.html | 12 +- ....Categories.Presheaf.NonPresheaf.Free.html | 24 +- Cubical.Categories.Presheaf.Properties.html | 192 +++--- ...cal.Categories.Presheaf.Representable.html | 20 +- Cubical.Categories.Profunctor.Base.html | 66 +- Cubical.Categories.RezkCompletion.Base.html | 4 +- ...ategories.RezkCompletion.Construction.html | 2 +- ...gories.TypesOfCategories.TypeCategory.html | 60 +- Cubical.Categories.UnderlyingGraph.html | 16 +- Cubical.Categories.Yoneda.html | 130 ++-- Cubical.Codata.M.AsLimit.M.Base.html | 6 +- Cubical.Cohomology.Base.html | 2 +- ...gy.EilenbergMacLane.EilenbergSteenrod.html | 12 +- ...Cohomology.EilenbergMacLane.Groups.Sn.html | 2 +- ...cal.Cohomology.EilenbergMacLane.Gysin.html | 2 +- ...mology.EilenbergMacLane.RingStructure.html | 6 +- Cubical.Data.Cardinality.Properties.html | 36 +- Cubical.Data.Fin.Arithmetic.html | 8 +- Cubical.Data.Fin.Base.html | 4 +- Cubical.Data.Fin.LehmerCode.html | 12 +- Cubical.Data.Fin.Properties.html | 18 +- Cubical.Data.FinSet.Base.html | 6 +- Cubical.Data.FinSet.Binary.Large.html | 2 +- Cubical.Data.FinSet.Cardinality.html | 6 +- Cubical.Data.FinSet.Constructors.html | 6 +- Cubical.Data.FinSet.DecidablePredicate.html | 10 +- Cubical.Data.FinSet.FiniteChoice.html | 6 +- Cubical.Data.FinSet.Quotients.html | 2 +- Cubical.Data.FinType.FiniteStructure.html | 2 +- Cubical.Data.FinType.Sigma.html | 4 +- Cubical.Data.Int.Divisibility.html | 4 +- Cubical.Data.Int.Order.html | 2 +- Cubical.Data.Nat.Divisibility.html | 2 +- Cubical.Data.Nat.GCD.html | 2 +- Cubical.Data.Nat.Order.html | 4 +- Cubical.Data.Prod.Properties.html | 2 +- ...a.Rationals.MoreRationals.SigmaQ.Base.html | 4 +- ...onals.MoreRationals.SigmaQ.Properties.html | 6 +- Cubical.Data.Rationals.Order.html | 4 +- Cubical.Data.Sigma.Properties.html | 599 ++++++++--------- Cubical.Data.SumFin.Properties.html | 20 +- Cubical.Displayed.Base.html | 2 +- Cubical.Displayed.Prop.html | 4 +- Cubical.Displayed.Properties.html | 2 +- Cubical.Displayed.Sigma.html | 6 +- Cubical.Experiments.EscardoSIP.html | 6 +- Cubical.Experiments.Poset.html | 12 +- ...periments.ZCohomologyOld.KcompPrelims.html | 2 +- Cubical.Foundations.Cubes.HLevels.html | 4 +- Cubical.Foundations.Cubes.Subtypes.html | 2 +- Cubical.Foundations.Equiv.Properties.html | 6 +- Cubical.Foundations.HLevels.html | 20 +- Cubical.Foundations.Pointed.Homotopy.html | 2 +- Cubical.Foundations.SIP.html | 2 +- Cubical.Functions.Embedding.html | 28 +- Cubical.Functions.Image.html | 4 +- Cubical.HITs.Cost.Base.html | 2 +- ...l.HITs.CumulativeHierarchy.Properties.html | 10 +- Cubical.HITs.FreeGroup.Properties.html | 385 +++++------ Cubical.HITs.FreeGroupoid.Properties.html | 18 +- ...l.HITs.James.Inductive.PushoutFormula.html | 2 +- Cubical.HITs.James.LoopSuspEquiv.html | 4 +- ...Ts.PropositionalTruncation.Properties.html | 2 +- Cubical.HITs.Pushout.KrausVonRaumer.html | 2 +- Cubical.HITs.RPn.Base.html | 14 +- Cubical.HITs.SetQuotients.Properties.html | 4 +- Cubical.HITs.SetTruncation.Fibers.html | 2 +- Cubical.HITs.SmashProduct.Base.html | 10 +- Cubical.Homotopy.BlakersMassey.html | 16 +- Cubical.Homotopy.Connected.html | 6 +- ....Homotopy.EilenbergMacLane.CupProduct.html | 4 +- ...opy.EilenbergMacLane.GradedCommTensor.html | 2 +- Cubical.Homotopy.EilenbergMacLane.Order2.html | 2 +- ....Homotopy.EilenbergMacLane.Properties.html | 8 +- Cubical.Homotopy.Group.Base.html | 2 +- Cubical.Homotopy.Group.LES.html | 4 +- ...l.Homotopy.Group.Pi4S3.BrunerieNumber.html | 12 +- Cubical.Homotopy.Group.Pi4S3.DirectProof.html | 2 +- Cubical.Homotopy.HopfInvariant.Brunerie.html | 4 +- Cubical.Homotopy.Whitehead.html | 4 +- Cubical.Papers.AffineSchemes.html | 2 +- ...elation.Nullary.DecidablePropositions.html | 2 +- Cubical.Structures.Axioms.html | 2 +- Cubical.Structures.Product.html | 4 +- Cubical.Structures.Relational.Equalizer.html | 4 +- Cubical.Structures.Relational.Function.html | 2 +- Cubical.Structures.Relational.Product.html | 6 +- Cubical.Tactics.CategorySolver.Solver.html | 12 +- Cubical.Tactics.FunctorSolver.Examples.html | 14 +- Cubical.Tactics.FunctorSolver.Reflection.html | 4 +- Cubical.Tactics.FunctorSolver.Solver.html | 26 +- Cubical.ZCohomology.EilenbergSteenrodZ.html | 22 +- Cubical.ZCohomology.GroupStructure.html | 12 +- Cubical.ZCohomology.Groups.CP2.html | 2 +- Cubical.ZCohomology.Groups.KleinBottle.html | 14 +- Cubical.ZCohomology.Groups.Prelims.html | 2 +- Cubical.ZCohomology.Groups.RP2.html | 6 +- Cubical.ZCohomology.Groups.Torus.html | 4 +- Cubical.ZCohomology.Groups.Wedge.html | 10 +- Cubical.ZCohomology.Gysin.html | 2 +- Cubical.ZCohomology.Properties.html | 4 +- ...homology.RingStructure.CohomologyRing.html | 6 +- ...ology.RingStructure.CohomologyRingFun.html | 6 +- 222 files changed, 4170 insertions(+), 3871 deletions(-) create mode 100644 Cubical.Categories.Instances.Groups.html diff --git a/Cubical.Algebra.AbGroup.Base.html b/Cubical.Algebra.AbGroup.Base.html index 39a6514e42..8fd9c05aa7 100644 --- a/Cubical.Algebra.AbGroup.Base.html +++ b/Cubical.Algebra.AbGroup.Base.html @@ -346,9 +346,9 @@ isAbGroup (snd HomGroup) = makeIsAbGroup isSetGroupHom - (λ { (f , p) (g , q) (h , r) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) + (λ { (f , p) (g , q) (h , r) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ x → +AssocB _ _ _) }) - (λ { (f , p) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ y → +IdRB _)}) - ((λ { (f , p) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ y → +InvRB _)})) - (λ { (f , p) (g , q) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ x → +CommB _ _)}) + (λ { (f , p) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ y → +IdRB _)}) + ((λ { (f , p) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ y → +InvRB _)})) + (λ { (f , p) (g , q) → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ x → +CommB _ _)}) \ No newline at end of file diff --git a/Cubical.Algebra.AbGroup.TensorProduct.html b/Cubical.Algebra.AbGroup.TensorProduct.html index 14d4f9c293..9d39468ca8 100644 --- a/Cubical.Algebra.AbGroup.TensorProduct.html +++ b/Cubical.Algebra.AbGroup.TensorProduct.html @@ -298,7 +298,7 @@ ∙ IsGroupHom.pres· q _ _ snd (tensorFun A B T C (f , p) (g , q)) = makeIsGroupHom - λ x y → Σ≡Prop (λ _ → isPropIsGroupHom _ _) + λ x y → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ b → cong g (funExt⁻ (cong fst (IsGroupHom.pres· p x y)) b) ∙ IsGroupHom.pres· q _ _) @@ -333,7 +333,7 @@ fst (fst mainF a) b = a ⊗ b snd (fst mainF a) = makeIsGroupHom (⊗DistR+⊗ a) snd mainF = makeIsGroupHom - λ x y → Σ≡Prop (λ _ → isPropIsGroupHom _ _) + λ x y → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (⊗DistL+⊗ x y)) isTensorProduct⨂ : (isTensorProductOf AGr and BGr) (AGr ⨂ BGr) @@ -361,10 +361,10 @@ (GroupHom (AGr *) (HomGroup (BGr *) C *)) Iso.fun mainIso = _ Iso.inv mainIso = invF - Iso.rightInv mainIso (f , p) = Σ≡Prop (λ _ → isPropIsGroupHom _ _) - (funExt λ a → Σ≡Prop (λ _ → isPropIsGroupHom _ _) refl) + Iso.rightInv mainIso (f , p) = Σ≡Prop (λ _ → isPropIsGroupHom _ _) + (funExt λ a → Σ≡Prop (λ _ → isPropIsGroupHom _ _) refl) Iso.leftInv mainIso (f , p) = - Σ≡Prop (λ _ → isPropIsGroupHom _ _) + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (⊗elimProp (λ _ → is-set (snd C) _ _) (λ _ _ → refl) λ x y ind1 ind2 → cong₂ (_+G_ (snd C)) ind1 ind2 @@ -537,7 +537,7 @@ ((compGroupHom (lIncl⨂ (RingStr.1r (snd G))) TensorMultHom)) idGroupHom G→G⨂G→Gₗ {G = G} = - Σ≡Prop (λ _ → isPropIsGroupHom _ _) + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (RingStr.·IdR (snd G))) G→G⨂G→Gᵣ : {G : Ring ℓ} @@ -545,6 +545,6 @@ ((compGroupHom (rIncl⨂ (RingStr.1r (snd G))) TensorMultHom)) idGroupHom G→G⨂G→Gᵣ {G = G} = - Σ≡Prop (λ _ → isPropIsGroupHom _ _) + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (RingStr.·IdL (snd G))) \ No newline at end of file diff --git a/Cubical.Algebra.Algebra.Base.html b/Cubical.Algebra.Algebra.Base.html index 9c49fd641a..585c6cc046 100644 --- a/Cubical.Algebra.Algebra.Base.html +++ b/Cubical.Algebra.Algebra.Base.html @@ -236,7 +236,7 @@ module N = AlgebraStr (str N) AlgebraHom≡ : {φ ψ : AlgebraHom A B} → fst φ ≡ fst ψ → φ ≡ ψ -AlgebraHom≡ = Σ≡Prop λ f → isPropIsAlgebraHom _ _ f _ +AlgebraHom≡ = Σ≡Prop λ f → isPropIsAlgebraHom _ _ f _ 𝒮ᴰ-Algebra : (R : Ring ℓ) → DUARel (𝒮-Univ ℓ') (AlgebraStr R) (ℓ-max ℓ ℓ') 𝒮ᴰ-Algebra R = diff --git a/Cubical.Algebra.Algebra.Properties.html b/Cubical.Algebra.Algebra.Properties.html index 78a7c1636c..5795f9d26e 100644 --- a/Cubical.Algebra.Algebra.Properties.html +++ b/Cubical.Algebra.Algebra.Properties.html @@ -178,11 +178,11 @@ fun isoOnHoms g = g ∘a AlgebraEquiv→AlgebraHom f inv isoOnHoms h = h ∘a AlgebraEquiv→AlgebraHom f⁻¹ rightInv isoOnHoms h = - Σ≡Prop + Σ≡Prop (λ h → isPropIsAlgebraHom _ (A .snd) h (C .snd)) (isoOnTypes .rightInv (h .fst)) leftInv isoOnHoms g = - Σ≡Prop + Σ≡Prop (λ g → isPropIsAlgebraHom _ (B .snd) g (C .snd)) (isoOnTypes .leftInv (g .fst)) @@ -221,7 +221,7 @@ ∙∙ transportTransport⁻ (ua (Algebra≡ A B)) q where helper : transport (sym (ua (Algebra≡ A B))) p ≡ transport (sym (ua (Algebra≡ A B))) q - helper = Σ≡Prop + helper = Σ≡Prop (λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) diff --git a/Cubical.Algebra.Algebra.Subalgebra.html b/Cubical.Algebra.Algebra.Subalgebra.html index 41eeeede89..93accc51c5 100644 --- a/Cubical.Algebra.Algebra.Subalgebra.html +++ b/Cubical.Algebra.Algebra.Subalgebra.html @@ -57,7 +57,7 @@ private module algStr = AlgebraStr Subalgebra→Algebra≡ : {x y : Σ[ a ∈ ⟨ A ⟩ ] a ∈ S} → fst x ≡ fst y → x ≡ y - Subalgebra→Algebra≡ eq = Σ≡Prop (∈-isProp S) eq + Subalgebra→Algebra≡ eq = Σ≡Prop (∈-isProp S) eq Subalgebra→Algebra : Algebra R ℓ' Subalgebra→Algebra .fst = Σ[ a ∈ ⟨ A ⟩ ] a ∈ S diff --git a/Cubical.Algebra.CommAlgebra.FPAlgebra.Base.html b/Cubical.Algebra.CommAlgebra.FPAlgebra.Base.html index 8d77a7a21b..fe44a521db 100644 --- a/Cubical.Algebra.CommAlgebra.FPAlgebra.Base.html +++ b/Cubical.Algebra.CommAlgebra.FPAlgebra.Base.html @@ -170,7 +170,7 @@ universal = (inducedHom , inducedHomOnGenerators) , λ {(f , mapsValues) - → Σ≡Prop (λ _ → isPropΠ (λ _ → is-set _ _)) + → Σ≡Prop (λ _ → isPropΠ (λ _ → is-set _ _)) (unique f mapsValues)} where open CommAlgebraStr (str A) @@ -216,7 +216,7 @@ Iso.fun FPHomIso = evaluateAtFP Iso.inv FPHomIso = inducedHomFP _ Iso.rightInv (FPHomIso {A}) = - λ b → Σ≡Prop + λ b → Σ≡Prop (λ x → isPropΠ (λ i → is-set (evPoly A (relation i) x) @@ -225,7 +225,7 @@ where open CommAlgebraStr (str A) Iso.leftInv (FPHomIso {A}) = - λ a → Σ≡Prop (λ f → isPropIsCommAlgebraHom {ℓ} {R} {ℓ} {ℓ} {FPAlgebra} {A} f) + λ a → Σ≡Prop (λ f → isPropIsCommAlgebraHom {ℓ} {R} {ℓ} {ℓ} {FPAlgebra} {A} f) λ i → fst (unique A (fst (evaluateAtFP {A} a)) (snd (evaluateAtFP a)) diff --git a/Cubical.Algebra.CommAlgebra.FreeCommAlgebra.Properties.html b/Cubical.Algebra.CommAlgebra.FreeCommAlgebra.Properties.html index c30025984b..671b3a49aa 100644 --- a/Cubical.Algebra.CommAlgebra.FreeCommAlgebra.Properties.html +++ b/Cubical.Algebra.CommAlgebra.FreeCommAlgebra.Properties.html @@ -27,7 +27,7 @@ open import Cubical.Foundations.Function hiding (const) open import Cubical.Foundations.Isomorphism -open import Cubical.Data.Sigma.Properties using (Σ≡Prop) +open import Cubical.Data.Sigma.Properties using (Σ≡Prop) open import Cubical.HITs.SetTruncation open import Cubical.Algebra.CommRing @@ -282,7 +282,7 @@ Iso.inv (homMapIso A) = inducedHom A Iso.rightInv (homMapIso A) = λ ϕ → Theory.mapRetrievable A ϕ Iso.leftInv (homMapIso {R = R} {I = I} A) = - λ f → Σ≡Prop (λ f → isPropIsCommAlgebraHom {M = R [ I ]} {N = A} f) + λ f → Σ≡Prop (λ f → isPropIsCommAlgebraHom {M = R [ I ]} {N = A} f) (Theory.homRetrievable A f) inducedHomUnique : @@ -379,7 +379,7 @@ from-to : (x : _) → from (to x) ≡ x from-to x = - Σ≡Prop (λ f → isPropIsCommAlgebraHom {M = R [ ⊥ ]} {N = B} f) + Σ≡Prop (λ f → isPropIsCommAlgebraHom {M = R [ ⊥ ]} {N = B} f) (Theory.homRetrievable B x) equiv : CommAlgebraHom (R [ ⊥ ]) B ≃ (⊥ → fst B) diff --git a/Cubical.Algebra.CommAlgebra.Instances.Initial.html b/Cubical.Algebra.CommAlgebra.Instances.Initial.html index 45489d7dd8..9ac4ddc656 100644 --- a/Cubical.Algebra.CommAlgebra.Instances.Initial.html +++ b/Cubical.Algebra.CommAlgebra.Instances.Initial.html @@ -7,7 +7,7 @@ open import Cubical.Foundations.Isomorphism open import Cubical.Data.Unit -open import Cubical.Data.Sigma.Properties using (Σ≡Prop) +open import Cubical.Data.Sigma.Properties using (Σ≡Prop) open import Cubical.Algebra.CommRing open import Cubical.Algebra.Ring @@ -72,7 +72,7 @@ → f ≡ initialMap initialMapEq f = let open IsAlgebraHom (snd f) - in Σ≡Prop + in Σ≡Prop (isPropIsCommAlgebraHom {M = initialCAlg} {N = A}) λ i x → ((fst f) x ≡⟨ cong (fst f) (sym (·IdR _)) ⟩ diff --git a/Cubical.Algebra.CommAlgebra.Instances.Unit.html b/Cubical.Algebra.CommAlgebra.Instances.Unit.html index b6d28bc316..de563c0a9d 100644 --- a/Cubical.Algebra.CommAlgebra.Instances.Unit.html +++ b/Cubical.Algebra.CommAlgebra.Instances.Unit.html @@ -7,7 +7,7 @@ open import Cubical.Foundations.Structure open import Cubical.Data.Unit -open import Cubical.Data.Sigma.Properties using (Σ≡Prop) +open import Cubical.Data.Sigma.Properties using (Σ≡Prop) open import Cubical.Algebra.CommRing open import Cubical.Algebra.CommAlgebra.Base @@ -42,7 +42,7 @@ terminalityContr : isContr (CommAlgebraHom A UnitCommAlgebra) terminalityContr = terminalMap , path where path : (ϕ : CommAlgebraHom A UnitCommAlgebra) → terminalMap ≡ ϕ - path ϕ = Σ≡Prop (isPropIsCommAlgebraHom {M = A} {N = UnitCommAlgebra}) + path ϕ = Σ≡Prop (isPropIsCommAlgebraHom {M = A} {N = UnitCommAlgebra}) λ i _ → isPropUnit* _ _ i open CommAlgebraStr (snd A) diff --git a/Cubical.Algebra.CommAlgebra.Localisation.html b/Cubical.Algebra.CommAlgebra.Localisation.html index 54376ca5d5..a8fdcd6e19 100644 --- a/Cubical.Algebra.CommAlgebra.Localisation.html +++ b/Cubical.Algebra.CommAlgebra.Localisation.html @@ -73,7 +73,7 @@ S⋆1⊆S⁻¹Rˣ : ∀ s → s ∈ S' → _⋆_ (snd S⁻¹RAsCommAlg) s (1a (snd S⁻¹RAsCommAlg)) ∈ S⁻¹Rˣ S⋆1⊆S⁻¹Rˣ s s∈S' = subst-∈ S⁻¹Rˣ - (cong [_] (≡-× (sym (·rRid s)) (Σ≡Prop (λ x → S' x .snd) (sym (·rRid _))))) + (cong [_] (≡-× (sym (·rRid s)) (Σ≡Prop (λ x → S' x .snd) (sym (·rRid _))))) (S/1⊆S⁻¹Rˣ s s∈S') @@ -109,7 +109,7 @@ χᴬuniqueness : (ψ : CommAlgebraHom S⁻¹RAsCommAlg B') → χᴬ ≡ ψ - χᴬuniqueness ψ = Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) + χᴬuniqueness ψ = Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) (cong (fst ∘ fst) (χuniqueness (ψ' , funExt ψ'r/1≡φr))) where χuniqueness = S⁻¹RHasUniversalProp B φ S⋆1⊆Bˣ .snd @@ -228,7 +228,7 @@ pres⋆ (snd center) = pres⋆ (snd χ₁) uniqueness : (φ : CommAlgebraEquiv S₁⁻¹RAsCommAlg S₂⁻¹RAsCommAlg) → center ≡ φ - uniqueness φ = Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) + uniqueness φ = Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) (equivEq (cong fst (S₁⁻¹RHasAlgUniversalProp S₂⁻¹RAsCommAlg S₁⊆S₂⁻¹Rˣ .snd (AlgebraEquiv→AlgebraHom φ)))) diff --git a/Cubical.Algebra.CommAlgebra.LocalisationAlgebra.html b/Cubical.Algebra.CommAlgebra.LocalisationAlgebra.html index c5654332c3..b3f3fc0643 100644 --- a/Cubical.Algebra.CommAlgebra.LocalisationAlgebra.html +++ b/Cubical.Algebra.CommAlgebra.LocalisationAlgebra.html @@ -163,7 +163,7 @@ (λ i → CommRingHom Aᵣ (S⁻¹AAsCommAlgebra→CommRing i)) (CommAlgebraHom→CommRingHom A S⁻¹AAsCommAlgebra /1AsCommAlgebraHom) RUniv./1AsCommRingHom - /1AsCommAlgebraHom→CommRingHom = ΣPathPProp (λ f → isPropIsRingHom _ f _) + /1AsCommAlgebraHom→CommRingHom = ΣPathPProp (λ f → isPropIsRingHom _ f _) (λ i → RUniv._/1) S⁻¹AHasUniversalProp : hasLocUniversalProp S⁻¹AAsCommAlgebra @@ -253,7 +253,7 @@ (fst φ) ∘ RUniv._/1 ≡ (fst ψ)) → (χₐ , χₐcomm) ≡ el χₐunique (φ' , φ'comm) = - Σ≡Prop ((λ _ → isSetΠ (λ _ → is-set) _ _)) $ AlgebraHom≡ $ + Σ≡Prop ((λ _ → isSetΠ (λ _ → is-set) _ _)) $ AlgebraHom≡ $ cong (fst ∘ fst) -- Get underlying bare function. (univ .snd (CommAlgebraHom→RingHom {A = S⁻¹AAsCommAlgebra} {B = B} φ' , φ'comm)) diff --git a/Cubical.Algebra.CommAlgebra.Properties.html b/Cubical.Algebra.CommAlgebra.Properties.html index e8e642dff5..d6a90eeb87 100644 --- a/Cubical.Algebra.CommAlgebra.Properties.html +++ b/Cubical.Algebra.CommAlgebra.Properties.html @@ -266,7 +266,7 @@ ∙∙ transportTransport⁻ (ua (CommAlgebra≡ A B)) q where helper : transport (sym (ua (CommAlgebra≡ A B))) p ≡ transport (sym (ua (CommAlgebra≡ A B))) q - helper = Σ≡Prop + helper = Σ≡Prop (λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) @@ -307,8 +307,8 @@ ---------------------------------------------------------------------------- → ∀ x y → isContr (CommAlgebraEquiv (σ x) (σ y)) contrCommAlgebraHom→contrCommAlgebraEquiv σ contrHom x y = σEquiv , - λ e → Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) - (Σ≡Prop isPropIsEquiv (cong fst (contrHom _ _ .snd (CommAlgebraEquiv→CommAlgebraHom e)))) + λ e → Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) + (Σ≡Prop isPropIsEquiv (cong fst (contrHom _ _ .snd (CommAlgebraEquiv→CommAlgebraHom e)))) where open Iso χ₁ = contrHom x y .fst @@ -380,5 +380,5 @@ pbSliceHom : Σ[ k ∈ CommRingHom (CommAlgebra→CommRing A) (CommAlgebra→CommRing B) ] k ∘r ((snd homA) ∘r f) ≡ ((snd homB) ∘r f) - pbSliceHom = fst asSliceHom , Σ≡Prop (λ _ → isPropIsRingHom _ _ _) λ i x → fst ((snd asSliceHom) i) (fst f x) + pbSliceHom = fst asSliceHom , Σ≡Prop (λ _ → isPropIsRingHom _ _ _) λ i x → fst ((snd asSliceHom) i) (fst f x) \ No newline at end of file diff --git a/Cubical.Algebra.CommAlgebra.QuotientAlgebra.html b/Cubical.Algebra.CommAlgebra.QuotientAlgebra.html index 18c19c3502..672323b23c 100644 --- a/Cubical.Algebra.CommAlgebra.QuotientAlgebra.html +++ b/Cubical.Algebra.CommAlgebra.QuotientAlgebra.html @@ -10,7 +10,7 @@ open import Cubical.HITs.SetQuotients hiding (_/_) open import Cubical.Data.Unit -open import Cubical.Data.Sigma.Properties using (Σ≡Prop) +open import Cubical.Data.Sigma.Properties using (Σ≡Prop) open import Cubical.Algebra.CommRing import Cubical.Algebra.CommRing.Quotient as CommRing @@ -161,7 +161,7 @@ unfolding inducedHom quotientHom inducedHom∘quotientHom : inducedHom ∘a quotientHom A I ≡ ϕ - inducedHom∘quotientHom = Σ≡Prop (isPropIsCommAlgebraHom {M = A} {N = B}) (funExt (λ a → refl)) + inducedHom∘quotientHom = Σ≡Prop (isPropIsCommAlgebraHom {M = A} {N = B}) (funExt (λ a → refl)) opaque unfolding quotientHom @@ -170,7 +170,7 @@ → f ∘a (quotientHom A I) ≡ g ∘a (quotientHom A I) → f ≡ g injectivePrecomp B f g p = - Σ≡Prop + Σ≡Prop (λ h → isPropIsCommAlgebraHom {M = A / I} {N = B} h) (descendMapPath (fst f) (fst g) is-set λ x → λ i → fst (p i) x) @@ -226,7 +226,7 @@ kernel≡I : kernel A (A / I) π ≡ I kernel≡I = - kernel A (A / I) π ≡⟨ Σ≡Prop + kernel A (A / I) π ≡⟨ Σ≡Prop (isPropIsCommIdeal (CommAlgebra→CommRing A)) refl ⟩ _ ≡⟨ CommRing.kernel≡I {R = CommAlgebra→CommRing A} I ⟩ diff --git a/Cubical.Algebra.CommAlgebra.UnivariatePolyList.html b/Cubical.Algebra.CommAlgebra.UnivariatePolyList.html index 090104be0f..8eb846929c 100644 --- a/Cubical.Algebra.CommAlgebra.UnivariatePolyList.html +++ b/Cubical.Algebra.CommAlgebra.UnivariatePolyList.html @@ -196,7 +196,7 @@ → fst f X ≡ x → f ≡ inducedHom inducedHomUnique f fX≡x = - Σ≡Prop + Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) λ i p → pwEq p i where @@ -227,7 +227,7 @@ fst (fst (equiv-proof (snd indcuedHomEquivalence) x)) = inducedHom x snd (fst (equiv-proof (snd indcuedHomEquivalence) x)) = inducedMapGenerator x snd (equiv-proof (snd indcuedHomEquivalence) x) (g , gX≡x) = - Σ≡Prop (λ _ → is-set _ _) (sym (inducedHomUnique x g gX≡x)) + Σ≡Prop (λ _ → is-set _ _) (sym (inducedHomUnique x g gX≡x)) equalByUMP : (f g : AlgebraHom (CommAlgebra→Algebra ListPolyCommAlgebra) A) → fst f X ≡ fst g X diff --git a/Cubical.Algebra.CommMonoid.Properties.html b/Cubical.Algebra.CommMonoid.Properties.html index 1612230af5..57a892e99b 100644 --- a/Cubical.Algebra.CommMonoid.Properties.html +++ b/Cubical.Algebra.CommMonoid.Properties.html @@ -35,11 +35,11 @@ IsCommMonoid.isMonoid (CommMonoidStr.isCommMonoid (snd makeSubCommMonoid)) = makeIsMonoid (isOfHLevelΣ 2 is-set λ _ → isProp→isSet (snd (P _))) - (λ x y z → Σ≡Prop (λ _ → snd (P _)) (·Assoc (fst x) (fst y) (fst z))) - (λ x → Σ≡Prop (λ _ → snd (P _)) (·IdR (fst x))) - λ x → Σ≡Prop (λ _ → snd (P _)) (·IdL (fst x)) + (λ x y z → Σ≡Prop (λ _ → snd (P _)) (·Assoc (fst x) (fst y) (fst z))) + (λ x → Σ≡Prop (λ _ → snd (P _)) (·IdR (fst x))) + λ x → Σ≡Prop (λ _ → snd (P _)) (·IdL (fst x)) IsCommMonoid.·Comm (CommMonoidStr.isCommMonoid (snd makeSubCommMonoid)) = - λ x y → Σ≡Prop (λ _ → snd (P _)) (·Comm (fst x) (fst y)) + λ x y → Σ≡Prop (λ _ → snd (P _)) (·Comm (fst x) (fst y)) module CommMonoidTheory (M' : CommMonoid ℓ) where open CommMonoidStr (snd M') diff --git a/Cubical.Algebra.CommRing.Base.html b/Cubical.Algebra.CommRing.Base.html index 9545cee010..eee77af81d 100644 --- a/Cubical.Algebra.CommRing.Base.html +++ b/Cubical.Algebra.CommRing.Base.html @@ -166,7 +166,7 @@ fst (inv (CommRingEquivIsoCommRingIso R S) e) = isoToEquiv (e .fst) snd (inv (CommRingEquivIsoCommRingIso R S) e) = e .snd rightInv (CommRingEquivIsoCommRingIso R S) (e , he) = - Σ≡Prop (λ e → isPropIsRingHom (snd (CommRing→Ring R)) (e .fun) (snd (CommRing→Ring S))) + Σ≡Prop (λ e → isPropIsRingHom (snd (CommRing→Ring R)) (e .fun) (snd (CommRing→Ring S))) rem where rem : equivToIso (isoToEquiv e) ≡ e @@ -175,7 +175,7 @@ rightInv (rem i) b j = CommRingStr.is-set (snd S) (fun e (inv e b)) b (rightInv e b) (rightInv e b) i j leftInv (rem i) a j = CommRingStr.is-set (snd R) (inv e (fun e a)) a (retEq (isoToEquiv e) a) (leftInv e a) i j leftInv (CommRingEquivIsoCommRingIso R S) e = - Σ≡Prop (λ e → isPropIsRingHom (snd (CommRing→Ring R)) (e .fst) (snd (CommRing→Ring S))) + Σ≡Prop (λ e → isPropIsRingHom (snd (CommRing→Ring R)) (e .fst) (snd (CommRing→Ring S))) (equivEq refl) isGroupoidCommRing : isGroupoid (CommRing ℓ) diff --git a/Cubical.Algebra.CommRing.FiberedProduct.html b/Cubical.Algebra.CommRing.FiberedProduct.html index 1bce245ef0..43b7761cfd 100644 --- a/Cubical.Algebra.CommRing.FiberedProduct.html +++ b/Cubical.Algebra.CommRing.FiberedProduct.html @@ -31,7 +31,7 @@ fbT = Σ[ ab ∈ fst A × fst B ] (fst α (fst ab) ≡ fst β (snd ab)) fbT≡ : {x y : fbT} → fst (fst x) ≡ fst (fst y) → snd (fst x) ≡ snd (fst y) → x ≡ y - fbT≡ h1 h2 = Σ≡Prop (λ _ → is-set (snd C) _ _) λ i → (h1 i) , (h2 i) + fbT≡ h1 h2 = Σ≡Prop (λ _ → is-set (snd C) _ _) λ i → (h1 i) , (h2 i) 0fbT : fbT 0fbT = (A.0r , B.0r) , α.pres0 ∙ sym β.pres0 diff --git a/Cubical.Algebra.CommRing.Ideal.Base.html b/Cubical.Algebra.CommRing.Ideal.Base.html index 3626a7186e..6dcbe537db 100644 --- a/Cubical.Algebra.CommRing.Ideal.Base.html +++ b/Cubical.Algebra.CommRing.Ideal.Base.html @@ -79,7 +79,7 @@ subst-∈ I = subst-∈p (I .fst) CommIdeal≡Char : {I J : CommIdeal} → I ⊆ J → J ⊆ I → I ≡ J - CommIdeal≡Char I⊆J J⊆I = Σ≡Prop isPropIsCommIdeal (⊆-extensionality _ _ (I⊆J , J⊆I)) + CommIdeal≡Char I⊆J J⊆I = Σ≡Prop isPropIsCommIdeal (⊆-extensionality _ _ (I⊆J , J⊆I)) -Closed : (I : CommIdeal) (x : R) → x ∈ I → (- x) ∈ I diff --git a/Cubical.Algebra.CommRing.Localisation.Base.html b/Cubical.Algebra.CommRing.Localisation.Base.html index a028e4d75f..ce8476ce30 100644 --- a/Cubical.Algebra.CommRing.Localisation.Base.html +++ b/Cubical.Algebra.CommRing.Localisation.Base.html @@ -112,7 +112,7 @@ +ₚ-symm : (a b : R × S) → (a +ₚ b) ≡ (b +ₚ a) +ₚ-symm (r₁ , s₁ , s₁∈S) (r₂ , s₂ , s₂∈S) = - ΣPathP (+Comm _ _ , Σ≡Prop (λ x → S' x .snd) (·Comm _ _)) + ΣPathP (+Comm _ _ , Σ≡Prop (λ x → S' x .snd) (·Comm _ _)) θ : (a a' b : R × S) → a ≈ a' → (a +ₚ b) ≈ (a' +ₚ b) θ (r₁ , s₁ , s₁∈S) (r'₁ , s'₁ , s'₁∈S) (r₂ , s₂ , s₂∈S) ((s , s∈S) , p) = (s , s∈S) , path @@ -148,7 +148,7 @@ where +ₗ-assoc[] : (a b c : R × S) → [ a ] +ₗ ([ b ] +ₗ [ c ]) ≡ ([ a ] +ₗ [ b ]) +ₗ [ c ] +ₗ-assoc[] (r , s , s∈S) (r' , s' , s'∈S) (r'' , s'' , s''∈S) = - cong [_] (ΣPathP ((path r s r' s' r'' s'') , Σ≡Prop (λ x → ∈-isProp S' x) (·Assoc _ _ _))) + cong [_] (ΣPathP ((path r s r' s' r'' s'') , Σ≡Prop (λ x → ∈-isProp S' x) (·Assoc _ _ _))) where path : (r s r' s' r'' s'' : R) → r · (s' · s'') + (r' · s'' + r'' · s') · s ≡ (r · s' + r' · s) · s'' + r'' · (s · s') @@ -169,7 +169,7 @@ path : [ r · 1r + 0r · s , s · 1r , SMultClosedSubset .multClosed s∈S (SMultClosedSubset .containsOne) ] ≡ [ r , s , s∈S ] - path = cong [_] (ΣPathP (eq1 r s , Σ≡Prop (λ x → ∈-isProp S' x) (·IdR _))) + path = cong [_] (ΣPathP (eq1 r s , Σ≡Prop (λ x → ∈-isProp S' x) (·IdR _))) -ₗ_ : S⁻¹R → S⁻¹R -ₗ_ = SQ.rec squash/ -ₗ[] -ₗWellDef @@ -203,7 +203,7 @@ where +ₗ-comm[] : (a b : R × S) → ([ a ] +ₗ [ b ]) ≡ ([ b ] +ₗ [ a ]) +ₗ-comm[] (r , s , s∈S) (r' , s' , s'∈S) = - cong [_] (ΣPathP ((+Comm _ _) , Σ≡Prop (λ x → ∈-isProp S' x) (·Comm _ _))) + cong [_] (ΣPathP ((+Comm _ _) , Σ≡Prop (λ x → ∈-isProp S' x) (·Comm _ _))) -- Now for multiplication @@ -217,7 +217,7 @@ ·ₚ-symm : (a b : R × S) → (a ·ₚ b) ≡ (b ·ₚ a) ·ₚ-symm (r₁ , s₁ , s₁∈S) (r₂ , s₂ , s₂∈S) = - ΣPathP (·Comm _ _ , Σ≡Prop (λ x → S' x .snd) (·Comm _ _)) + ΣPathP (·Comm _ _ , Σ≡Prop (λ x → S' x .snd) (·Comm _ _)) θ : (a a' b : R × S) → a ≈ a' → (a ·ₚ b) ≈ (a' ·ₚ b) θ (r₁ , s₁ , s₁∈S) (r'₁ , s'₁ , s'₁∈S) (r₂ , s₂ , s₂∈S) ((s , s∈S) , p) = (s , s∈S) , path @@ -243,13 +243,13 @@ where ·ₗ-assoc[] : (a b c : R × S) → [ a ] ·ₗ ([ b ] ·ₗ [ c ]) ≡ ([ a ] ·ₗ [ b ]) ·ₗ [ c ] ·ₗ-assoc[] (r , s , s∈S) (r' , s' , s'∈S) (r'' , s'' , s''∈S) = - cong [_] (ΣPathP ((·Assoc _ _ _) , Σ≡Prop (λ x → ∈-isProp S' x) (·Assoc _ _ _))) + cong [_] (ΣPathP ((·Assoc _ _ _) , Σ≡Prop (λ x → ∈-isProp S' x) (·Assoc _ _ _))) ·ₗ-rid : (x : S⁻¹R) → x ·ₗ 1ₗ ≡ x ·ₗ-rid = SQ.elimProp (λ _ → squash/ _ _) ·ₗ-rid[] where ·ₗ-rid[] : (a : R × S) → ([ a ] ·ₗ 1ₗ) ≡ [ a ] - ·ₗ-rid[] (r , s , s∈S) = cong [_] (ΣPathP ((·IdR _) , Σ≡Prop (λ x → ∈-isProp S' x) (·IdR _))) + ·ₗ-rid[] (r , s , s∈S) = cong [_] (ΣPathP ((·IdR _) , Σ≡Prop (λ x → ∈-isProp S' x) (·IdR _))) ·ₗ-rdist-+ₗ : (x y z : S⁻¹R) → x ·ₗ (y +ₗ z) ≡ (x ·ₗ y) +ₗ (x ·ₗ z) @@ -270,7 +270,7 @@ where ·ₗ-comm[] : (a b : R × S) → [ a ] ·ₗ [ b ] ≡ [ b ] ·ₗ [ a ] ·ₗ-comm[] (r , s , s∈S) (r' , s' , s'∈S) = - cong [_] (ΣPathP ((·Comm _ _) , Σ≡Prop (λ x → ∈-isProp S' x) (·Comm _ _))) + cong [_] (ΣPathP ((·Comm _ _) , Σ≡Prop (λ x → ∈-isProp S' x) (·Comm _ _))) diff --git a/Cubical.Algebra.CommRing.Localisation.InvertingElements.html b/Cubical.Algebra.CommRing.Localisation.InvertingElements.html index 466dc67b40..ab9a2070e7 100644 --- a/Cubical.Algebra.CommRing.Localisation.InvertingElements.html +++ b/Cubical.Algebra.CommRing.Localisation.InvertingElements.html @@ -130,7 +130,7 @@ []-case (r , s , s∈S[f]) = PT.rec (PisProp _) Σhelper s∈S[f] where Σhelper : Σ[ n ∈ ℕ ] s ≡ f ^ n → P [ r , s , s∈S[f] ] - Σhelper (n , p) = subst P (cong [_] (≡-× refl (Σ≡Prop (λ _ → isPropPropTrunc) (sym p)))) (base r n) + Σhelper (n , p) = subst P (cong [_] (≡-× refl (Σ≡Prop (λ _ → isPropPropTrunc) (sym p)))) (base r n) invElPropElim2 : {f g : R} {P : R[1/ f ] → R[1/ g ] → Type ℓ'} @@ -390,16 +390,16 @@ (cong₂ _+_ (sym (·IdR _) ∙ (λ i → (·IdR r (~ i)) · (·IdR 1r (~ i)))) (sym (·IdR _) ∙ (λ i → (·IdR r' (~ i)) · (·IdR 1r (~ i))))) - (Σ≡Prop (λ _ → isPropPropTrunc) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdL _) ∙ (λ i → (·IdL 1r (~ i)) · (·IdL 1r (~ i))))))) - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·ᶠ-lid 1ᶠ)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·ᶠ-lid 1ᶠ)))) lem· : _ lem· r r' = cong [_] (≡-× - (cong [_] (≡-× refl (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdL _))))) - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·ᶠ-lid 1ᶠ)))) + (cong [_] (≡-× refl (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdL _))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·ᶠ-lid 1ᶠ)))) -- this will give us a map R[1/fg] → R[1/f][1/g] by the universal property of localisation fⁿgⁿ/1/1∈R[1/f][1/g]ˣ : (s : R) → s ∈ ([_ⁿ|n≥0] R' (f · g)) → s /1/1 ∈ R[1/f][1/g]ˣ @@ -467,7 +467,7 @@ path : [ g ^ n · r , 1r , PT.∣ 0 , refl ∣₁ ] ≡ 0ᶠ path = [ g ^ n · r , 1r , PT.∣ 0 , refl ∣₁ ] - ≡⟨ cong [_] (≡-× refl (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR _)))) ⟩ + ≡⟨ cong [_] (≡-× refl (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR _)))) ⟩ [ g ^ n , 1r , PT.∣ 0 , refl ∣₁ ] ·ᶠ r/1 @@ -633,7 +633,7 @@ , g/1 ^ᶠ n , PT.∣ n , refl ∣₁ ] ·R[1/f][1/g] (x .snd .fst /1/1) base-^ᶠ-helper r m n = subst (λ y → ∃[ x ∈ R × S[fg] ] (x .fst /1/1) ≡ [ [ r , f ^ m , PT.∣ m , refl ∣₁ ] , y ] ·R[1/f][1/g] (x .snd .fst /1/1)) - (Σ≡Prop (λ _ → isPropPropTrunc) (^-respects-/1 _ n)) (baseCase r m n) + (Σ≡Prop (λ _ → isPropPropTrunc) (^-respects-/1 _ n)) (baseCase r m n) indStep : (r : R[1/_] R' f) (n : ℕ) → ∃[ x ∈ R × S[fg] ] (x .fst /1/1) ≡ [ r , g/1 ^ᶠ n , PT.∣ n , refl ∣₁ ] ·R[1/f][1/g] (x .snd .fst /1/1) @@ -833,9 +833,9 @@ ℕcase : (r : R) (n : ℕ) → φ [ r , (f · g) ^ n , PT.∣ n , refl ∣₁ ] ≡ χ [ r , (f · g) ^ n , PT.∣ n , refl ∣₁ ] ℕcase r n = cong [_] (ΣPathP --look into the components of the double-fractions - ( cong [_] (ΣPathP (eq1 , Σ≡Prop (λ x → S'[f] x .snd) (sym (·IdL _)))) - , Σ≡Prop (λ x → S'[f][g] x .snd) --ignore proof that denominator is power of g/1 - ( cong [_] (ΣPathP (sym (·IdL _) , Σ≡Prop (λ x → S'[f] x .snd) (sym (·IdL _))))))) + ( cong [_] (ΣPathP (eq1 , Σ≡Prop (λ x → S'[f] x .snd) (sym (·IdL _)))) + , Σ≡Prop (λ x → S'[f][g] x .snd) --ignore proof that denominator is power of g/1 + ( cong [_] (ΣPathP (sym (·IdL _) , Σ≡Prop (λ x → S'[f] x .snd) (sym (·IdL _))))))) where S'[f] = ([_ⁿ|n≥0] R' f) S'[f][g] = ([_ⁿ|n≥0] R[1/f]AsCommRing [ g , 1r , powersFormMultClosedSubset R' f .containsOne ]) diff --git a/Cubical.Algebra.CommRing.Localisation.Limit.html b/Cubical.Algebra.CommRing.Localisation.Limit.html index 514c7ae229..450c83fc88 100644 --- a/Cubical.Algebra.CommRing.Localisation.Limit.html +++ b/Cubical.Algebra.CommRing.Localisation.Limit.html @@ -221,7 +221,7 @@ where χId : ∀ i x → χˡ i i .fst x ≡ χʳ i i .fst x χId i = invElPropElim (λ _ → squash/ _ _) - (λ r m → cong [_] (ΣPathP (refl , Σ≡Prop (∈-isProp _) refl))) + (λ r m → cong [_] (ΣPathP (refl , Σ≡Prop (∈-isProp _) refl))) aux (gt j<i) = χˡ i j .fst (x i) ≡⟨ χSwapL→R i j (x i) ⟩ @@ -234,14 +234,14 @@ χSwapL→R : ∀ i j x → χˡ i j .fst x ≡ χʳsubst i j x χSwapL→R i j = invElPropElim (λ _ → squash/ _ _) - λ r m → cong [_] (ΣPathP (sym (transportRefl _) , Σ≡Prop (∈-isProp _) + λ r m → cong [_] (ΣPathP (sym (transportRefl _) , Σ≡Prop (∈-isProp _) (sym (transportRefl _ ∙ cong (λ x → 1r · transport refl (x ^ m)) (·Comm _ _))))) χˡsubst : (i j : Fin (suc n)) → R[1/ f j ] → R[1/ f i · f j ] χˡsubst i j x = subst (λ r → R[1/ r ]) (·Comm (f j) (f i)) (χˡ j i .fst x) χSwapR→L : ∀ i j x → χʳ i j .fst x ≡ χˡsubst i j x χSwapR→L i j = invElPropElim (λ _ → squash/ _ _) - λ r m → cong [_] (ΣPathP (sym (transportRefl _) , Σ≡Prop (∈-isProp _) + λ r m → cong [_] (ΣPathP (sym (transportRefl _) , Σ≡Prop (∈-isProp _) (sym (transportRefl _ ∙ cong (λ x → 1r · transport refl (x ^ m)) (·Comm _ _))))) χ≡PropElim : {B : ((i : Fin (suc n)) → R[1/ f i ]) → Type ℓ''} (isPropB : ∀ {x} → isProp (B x)) @@ -433,7 +433,7 @@ Σhelper : Σ[ α ∈ FinVec R (suc n) ] 1r ≡ linearCombination R' α f²ᵐ⁺ˡ → ∃![ y ∈ R ] (∀ i → y /1ˢ ≡ [ r i , f i ^ m , ∣ m , refl ∣₁ ]) Σhelper (α , linCombi) = (z , z≡r/fᵐ) - , λ y' → Σ≡Prop (λ _ → isPropΠ (λ _ → squash/ _ _)) (unique _ (y' .snd)) + , λ y' → Σ≡Prop (λ _ → isPropΠ (λ _ → squash/ _ _)) (unique _ (y' .snd)) where z = ∑ λ i → α i · r i · f i ^ (m +ℕ l) @@ -536,18 +536,18 @@ -} open Category (CommRingsCategory {ℓ}) open Cone - open Functor - - locDiagram : Functor (DLShfDiag (suc n) ℓ) CommRingsCategory - F-ob locDiagram (sing i) = R[1/ f i ]AsCommRing - F-ob locDiagram (pair i j _) = R[1/ f i · f j ]AsCommRing - F-hom locDiagram idAr = idCommRingHom _ - F-hom locDiagram singPairL = χˡ _ _ - F-hom locDiagram singPairR = χʳ _ _ - F-id locDiagram = refl - F-seq locDiagram idAr _ = sym (⋆IdL _) - F-seq locDiagram singPairL idAr = sym (⋆IdR _) - F-seq locDiagram singPairR idAr = sym (⋆IdR _) + open Functor + + locDiagram : Functor (DLShfDiag (suc n) ℓ) CommRingsCategory + F-ob locDiagram (sing i) = R[1/ f i ]AsCommRing + F-ob locDiagram (pair i j _) = R[1/ f i · f j ]AsCommRing + F-hom locDiagram idAr = idCommRingHom _ + F-hom locDiagram singPairL = χˡ _ _ + F-hom locDiagram singPairR = χʳ _ _ + F-id locDiagram = refl + F-seq locDiagram idAr _ = sym (⋆IdL _) + F-seq locDiagram singPairL idAr = sym (⋆IdR _) + F-seq locDiagram singPairR idAr = sym (⋆IdR _) locCone : Cone locDiagram R' coneOut locCone (sing i) = U./1AsCommRingHom i @@ -610,7 +610,7 @@ coneOut cᴬ (pair i j i<j) ∎ ψUniqueness : (y : Σ[ θ ∈ CommRingHom A' R' ] isConeMor cᴬ locCone θ) → (ψ , isConeMorψ) ≡ y - ψUniqueness (θ , isConeMorθ) = Σ≡Prop (λ _ → isPropIsConeMor _ _ _) + ψUniqueness (θ , isConeMorθ) = Σ≡Prop (λ _ → isPropIsConeMor _ _ _) (RingHom≡ (funExt λ a → cong fst (applyEqualizerLemma a .snd (θtriple a)))) where θtriple : (a : A) → Σ[ x ∈ R ] ∀ i → x /1ˢ ≡ φ i .fst a diff --git a/Cubical.Algebra.CommRing.Localisation.PullbackSquare.html b/Cubical.Algebra.CommRing.Localisation.PullbackSquare.html index a1727874fd..8cc1eca4c2 100644 --- a/Cubical.Algebra.CommRing.Localisation.PullbackSquare.html +++ b/Cubical.Algebra.CommRing.Localisation.PullbackSquare.html @@ -575,7 +575,7 @@ χUniqueness : (y : Σ[ θ ∈ CommRingHom A R' ] (ψ ≡ θ ⋆ /1ᵍAsCommRingHom) × (φ ≡ θ ⋆ /1ᶠAsCommRingHom)) → (χ , χCoh) ≡ y - χUniqueness (θ , θCoh) = Σ≡Prop (λ _ → isProp× (isSetRingHom _ _ _ _) + χUniqueness (θ , θCoh) = Σ≡Prop (λ _ → isProp× (isSetRingHom _ _ _ _) (isSetRingHom _ _ _ _)) (RingHom≡ (funExt (λ a → cong fst (applyEqualizerLemma a .snd (θtriple a))))) where diff --git a/Cubical.Algebra.CommRing.Localisation.UniversalProperty.html b/Cubical.Algebra.CommRing.Localisation.UniversalProperty.html index 8a83a9d352..ef3faaa863 100644 --- a/Cubical.Algebra.CommRing.Localisation.UniversalProperty.html +++ b/Cubical.Algebra.CommRing.Localisation.UniversalProperty.html @@ -80,8 +80,8 @@ makeIsRingHom refl (λ r r' → cong [_] (≡-× (cong₂ (_+_) (sym (·IdR r)) (sym (·IdR r'))) - (Σ≡Prop (λ x → S' x .snd) (sym (·IdL 1r))))) - (λ _ _ → cong [_] (≡-× refl (Σ≡Prop (λ x → S' x .snd) (sym (·IdL 1r))))) + (Σ≡Prop (λ x → S' x .snd) (sym (·IdL 1r))))) + (λ _ _ → cong [_] (≡-× refl (Σ≡Prop (λ x → S' x .snd) (sym (·IdL 1r))))) S⁻¹Rˣ = S⁻¹RAsCommRing ˣ S/1⊆S⁻¹Rˣ : ∀ s → s ∈ S' → (s /1) ∈ S⁻¹Rˣ @@ -298,7 +298,7 @@ χunique : (y : Σ[ χ' ∈ CommRingHom S⁻¹RAsCommRing B' ] fst χ' ∘ _/1 ≡ ψ₀) → (χ , funExt χcomp) ≡ y - χunique (χ' , χ'/1≡ψ) = Σ≡Prop (λ x → isSetΠ (λ _ → Bset) _ _) (RingHom≡ fχ≡fχ') + χunique (χ' , χ'/1≡ψ) = Σ≡Prop (λ x → isSetΠ (λ _ → Bset) _ _) (RingHom≡ fχ≡fχ') where open CommRingHomTheory {A' = S⁻¹RAsCommRing} {B' = B'} χ' renaming (φ[x⁻¹]≡φ[x]⁻¹ to χ'[x⁻¹]≡χ'[x]⁻¹) @@ -321,7 +321,7 @@ ·ₗ-path : [ r , s , s∈S' ] ≡ [ r , 1r , SMultClosedSubset .containsOne ] ·ₗ [ 1r , s , s∈S' ] - ·ₗ-path = cong [_] (≡-× (sym (·IdR r)) (Σ≡Prop (λ x → S' x .snd) (sym (·IdL s)))) + ·ₗ-path = cong [_] (≡-× (sym (·IdR r)) (Σ≡Prop (λ x → S' x .snd) (sym (·IdL s)))) instancepath : ⦃ _ : ψ₀ s ∈ Bˣ ⦄ ⦃ _ : s /1 ∈ S⁻¹Rˣ ⦄ ⦃ _ : fst χ' (s /1) ∈ Bˣ ⦄ → ψ₀ r ·B ψ₀ s ⁻¹ ≡ fst χ' [ r , s , s∈S' ] diff --git a/Cubical.Algebra.CommRing.Properties.html b/Cubical.Algebra.CommRing.Properties.html index 1462732d48..94349b3d1a 100644 --- a/Cubical.Algebra.CommRing.Properties.html +++ b/Cubical.Algebra.CommRing.Properties.html @@ -40,7 +40,7 @@ private R = fst R' inverseUniqueness : (r : R) → isProp (Σ[ r' ∈ R ] r · r' ≡ 1r) - inverseUniqueness r (r' , rr'≡1) (r'' , rr''≡1) = Σ≡Prop (λ _ → is-set _ _) path + inverseUniqueness r (r' , rr'≡1) (r'' , rr''≡1) = Σ≡Prop (λ _ → is-set _ _) path where path : r' ≡ r'' path = r' ≡⟨ sym (·IdR _) ⟩ @@ -329,7 +329,7 @@ ∙∙ transportTransport⁻ (ua (CommRing≡ A B)) q where helper : transport (sym (ua (CommRing≡ A B))) p ≡ transport (sym (ua (CommRing≡ A B))) q - helper = Σ≡Prop + helper = Σ≡Prop (λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) diff --git a/Cubical.Algebra.CommSemiring.Instances.UpperNat.html b/Cubical.Algebra.CommSemiring.Instances.UpperNat.html index a95e4bf70a..bcd9668349 100644 --- a/Cubical.Algebra.CommSemiring.Instances.UpperNat.html +++ b/Cubical.Algebra.CommSemiring.Instances.UpperNat.html @@ -173,7 +173,7 @@ AnnihilL : (x : ℕ↑b) → 0↑ · x ≡ 0↑ AnnihilL x = - Σ≡Prop (λ s → PropCompletion.isPropIsBounded ℓ-zero ℕ≤+ s) + Σ≡Prop (λ s → PropCompletion.isPropIsBounded ℓ-zero ℕ≤+ s) (pathFromImplications (fst (0↑ · x)) (fst 0↑) (⇒) ⇐) where ⇒ : (n : ℕ) → typeAt n (fst (0↑ · x)) → typeAt n (fst 0↑) @@ -205,7 +205,7 @@ (is-set +IsCM) (·Assoc +IsCM) (·IdR +IsCM) (·Comm +IsCM) (·Assoc ·IsCM) (·IdR ·IsCM) - (λ x y z → Σ≡Prop (λ s → PropCompletion.isPropIsBounded ℓ-zero ℕ≤+ s) + (λ x y z → Σ≡Prop (λ s → PropCompletion.isPropIsBounded ℓ-zero ℕ≤+ s) (ConstructionUnbounded.+LDist· (fst x) (fst y) (fst z))) AnnihilL (·Comm ·IsCM) diff --git a/Cubical.Algebra.DirectSum.DirectSumFun.Properties.html b/Cubical.Algebra.DirectSum.DirectSumFun.Properties.html index b49ffe5f00..ef828ef51a 100644 --- a/Cubical.Algebra.DirectSum.DirectSumFun.Properties.html +++ b/Cubical.Algebra.DirectSum.DirectSumFun.Properties.html @@ -66,18 +66,18 @@ -- AbGroup Properties +⊕FunAssoc : (x y z : ⊕Fun G Gstr) → x +⊕Fun (y +⊕Fun z) ≡ (x +⊕Fun y) +⊕Fun z +⊕FunAssoc (f , Anf) (g , Ang) (h , Anh) = - ΣPathTransport→PathΣ _ _ + ΣPathTransport→PathΣ _ _ (funExt (λ n → Gstr n .+Assoc _ _ _) , (squash₁ _ _)) +⊕FunRid : (x : ⊕Fun G Gstr) → x +⊕Fun 0⊕Fun ≡ x - +⊕FunRid (f , Anf) = ΣPathTransport→PathΣ _ _ + +⊕FunRid (f , Anf) = ΣPathTransport→PathΣ _ _ ((funExt (λ n → +IdR (Gstr n) _)) , squash₁ _ _) +⊕FunInvR : (x : ⊕Fun G Gstr) → x +⊕Fun Inv⊕Fun x ≡ 0⊕Fun - +⊕FunInvR (f , Anf) = ΣPathTransport→PathΣ _ _ + +⊕FunInvR (f , Anf) = ΣPathTransport→PathΣ _ _ ((funExt (λ n → +InvR (Gstr n) _)) , (squash₁ _ _)) +⊕FunComm : (x y : ⊕Fun G Gstr) → x +⊕Fun y ≡ y +⊕Fun x - +⊕FunComm (f , Anf) (g , Ang) = ΣPathTransport→PathΣ _ _ + +⊕FunComm (f , Anf) (g , Ang) = ΣPathTransport→PathΣ _ _ ((funExt (λ n → Gstr n .+Comm _ _)) , (squash₁ _ _)) \ No newline at end of file diff --git a/Cubical.Algebra.DirectSum.DirectSumHIT.UniversalProperty.html b/Cubical.Algebra.DirectSum.DirectSumHIT.UniversalProperty.html index ce5e023eca..d083012464 100644 --- a/Cubical.Algebra.DirectSum.DirectSumHIT.UniversalProperty.html +++ b/Cubical.Algebra.DirectSum.DirectSumHIT.UniversalProperty.html @@ -60,7 +60,7 @@ -- Universal Property up∃⊕HIT : (k : Idx) → fHhom k ≡ compGroupHom (injₖ-hom k) ⊕HIT→H-hom - up∃⊕HIT k = ΣPathTransport→PathΣ _ _ + up∃⊕HIT k = ΣPathTransport→PathΣ _ _ ((funExt (λ _ → refl)) , isPropIsGroupHom _ _ _ _) @@ -68,7 +68,7 @@ upUnicity⊕HIT : (hhom : AbGroupHom (⊕HIT-AbGr Idx G Gstr) HAbGr) → (eqInj : (k : Idx) → fHhom k ≡ compGroupHom (injₖ-hom k) hhom) → hhom ≡ ⊕HIT→H-hom - upUnicity⊕HIT (h , hstr) eqInj = ΣPathTransport→PathΣ _ _ + upUnicity⊕HIT (h , hstr) eqInj = ΣPathTransport→PathΣ _ _ (helper , isPropIsGroupHom _ _ _ _) where diff --git a/Cubical.Algebra.DirectSum.Equiv-DSHIT-DSFun.html b/Cubical.Algebra.DirectSum.Equiv-DSHIT-DSFun.html index f978f26ac0..bf10eed625 100644 --- a/Cubical.Algebra.DirectSum.Equiv-DSHIT-DSFun.html +++ b/Cubical.Algebra.DirectSum.Equiv-DSHIT-DSFun.html @@ -188,7 +188,7 @@ ⊕HIT→⊕Fun-pres0 = refl ⊕HIT→⊕Fun-pres+ : (x y : ⊕HIT ℕ G Gstr) → ⊕HIT→⊕Fun (x +⊕HIT y) ≡ ((⊕HIT→⊕Fun x) +⊕Fun (⊕HIT→⊕Fun y)) - ⊕HIT→⊕Fun-pres+ x y = ΣPathTransport→PathΣ _ _ (refl , (squash₁ _ _)) + ⊕HIT→⊕Fun-pres+ x y = ΣPathTransport→PathΣ _ _ (refl , (squash₁ _ _)) ----------------------------------------------------------------------------- @@ -272,10 +272,10 @@ ∙ sym q} inj-⊕HIT→⊕Fun : (x y : ⊕HIT ℕ G Gstr) → ⊕HIT→⊕Fun x ≡ ⊕HIT→⊕Fun y → x ≡ y - inj-⊕HIT→⊕Fun x y p = inj-⊕HIT→Fun x y (fst (PathΣ→ΣPathTransport _ _ p)) + inj-⊕HIT→⊕Fun x y p = inj-⊕HIT→Fun x y (fst (PathΣ→ΣPathTransport _ _ p)) lemProp : (g : ⊕Fun G Gstr) → isProp (Σ[ x ∈ ⊕HIT ℕ G Gstr ] ⊕HIT→⊕Fun x ≡ g ) - lemProp g (x , p) (y , q) = ΣPathTransport→PathΣ _ _ + lemProp g (x , p) (y , q) = ΣPathTransport→PathΣ _ _ ((inj-⊕HIT→⊕Fun x y (p ∙ sym q)) , isSet⊕Fun _ _ _ _) @@ -334,7 +334,7 @@ ⊕Fun→⊕HIT+ : (g : ⊕Fun G Gstr) → Σ[ x ∈ ⊕HIT ℕ G Gstr ] ⊕HIT→⊕Fun x ≡ g ⊕Fun→⊕HIT+ (g , Ang) = PT.rec (lemProp (g , Ang)) (λ { (k , ng) - → Strad g k , ΣPathTransport→PathΣ _ _ + → Strad g k , ΣPathTransport→PathΣ _ _ ((funExt (trad-section g k ng)) , (squash₁ _ _)) }) Ang where diff --git a/Cubical.Algebra.DistLattice.Basis.html b/Cubical.Algebra.DistLattice.Basis.html index f6753095be..b7a66d33aa 100644 --- a/Cubical.Algebra.DistLattice.Basis.html +++ b/Cubical.Algebra.DistLattice.Basis.html @@ -73,8 +73,8 @@ , isBasisS .∧lClosed _ _ (snd x) (snd y) isSemilattice (snd (Basis→MeetSemilattice S isBasisS)) = makeIsSemilattice (isSetΣ is-set λ _ → isProp→isSet (S _ .snd)) - (λ _ _ _ → Σ≡Prop (λ _ → S _ .snd) (∧lAssoc _ _ _)) - (λ _ → Σ≡Prop (λ _ → S _ .snd) (∧lRid _)) - (λ _ _ → Σ≡Prop (λ _ → S _ .snd) (∧lComm _ _)) - λ _ → Σ≡Prop (λ _ → S _ .snd) (∧lIdem _) + (λ _ _ _ → Σ≡Prop (λ _ → S _ .snd) (∧lAssoc _ _ _)) + (λ _ → Σ≡Prop (λ _ → S _ .snd) (∧lRid _)) + (λ _ _ → Σ≡Prop (λ _ → S _ .snd) (∧lComm _ _)) + λ _ → Σ≡Prop (λ _ → S _ .snd) (∧lIdem _) \ No newline at end of file diff --git a/Cubical.Algebra.GradedRing.DirectSumFun.html b/Cubical.Algebra.GradedRing.DirectSumFun.html index f6f7a72fef..d68605d5fc 100644 --- a/Cubical.Algebra.GradedRing.DirectSumFun.html +++ b/Cubical.Algebra.GradedRing.DirectSumFun.html @@ -132,7 +132,7 @@ ... | no ¬p = refl ⊕HIT→⊕Fun-pres1 : ⊕HIT→⊕Fun (base 0 1⋆) ≡ 1⊕Fun - ⊕HIT→⊕Fun-pres1 = ΣPathTransport→PathΣ _ _ + ⊕HIT→⊕Fun-pres1 = ΣPathTransport→PathΣ _ _ ((funExt (λ n → ⊕HIT→Fun-pres1 n)) , (squash₁ _ _)) ----------------------------------------------------------------------------- @@ -350,7 +350,7 @@ ⊕HIT→⊕Fun-pres-prodF : (x y : ⊕HIT ℕ G Gstr) → ⊕HIT→⊕Fun (x prod y) ≡ ((⊕HIT→⊕Fun x) prodF (⊕HIT→⊕Fun y)) - ⊕HIT→⊕Fun-pres-prodF x y = ΣPathTransport→PathΣ _ _ + ⊕HIT→⊕Fun-pres-prodF x y = ΣPathTransport→PathΣ _ _ ((⊕HIT→Fun-pres-prodFun x y) , (squash₁ _ _)) diff --git a/Cubical.Algebra.GradedRing.Instances.CohomologyRing.html b/Cubical.Algebra.GradedRing.Instances.CohomologyRing.html index 1626145cfe..d095675043 100644 --- a/Cubical.Algebra.GradedRing.Instances.CohomologyRing.html +++ b/Cubical.Algebra.GradedRing.Instances.CohomologyRing.html @@ -41,9 +41,9 @@ _⌣_ (λ {k} {l} → 0ₕ-⌣ k l) (λ {k} {l} → ⌣-0ₕ k l) - (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _)))) ) - (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) - (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) + (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _)))) ) + (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) + (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) (λ _ _ _ → leftDistr-⌣ _ _ _ _ _) ( λ _ _ _ → rightDistr-⌣ _ _ _ _ _) \ No newline at end of file diff --git a/Cubical.Algebra.GradedRing.Instances.CohomologyRingFun.html b/Cubical.Algebra.GradedRing.Instances.CohomologyRingFun.html index 574a729b4f..15d8e5adc3 100644 --- a/Cubical.Algebra.GradedRing.Instances.CohomologyRingFun.html +++ b/Cubical.Algebra.GradedRing.Instances.CohomologyRingFun.html @@ -40,9 +40,9 @@ _⌣_ (λ {k} {l} → 0ₕ-⌣ k l) (λ {k} {l} → ⌣-0ₕ k l) - (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _))))) - (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) - (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) + (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _))))) + (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) + (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) (λ _ _ _ → leftDistr-⌣ _ _ _ _ _) λ _ _ _ → rightDistr-⌣ _ _ _ _ _ \ No newline at end of file diff --git a/Cubical.Algebra.GradedRing.Instances.TrivialGradedRing.html b/Cubical.Algebra.GradedRing.Instances.TrivialGradedRing.html index 0a846e2d46..b3ab022d0c 100644 --- a/Cubical.Algebra.GradedRing.Instances.TrivialGradedRing.html +++ b/Cubical.Algebra.GradedRing.Instances.TrivialGradedRing.html @@ -62,18 +62,18 @@ _≡_ {A = Σ[ k ∈ ℕ ] G k} (k Cubical.Data.Nat.+ (l Cubical.Data.Nat.+ m) , ⋆ a (⋆ b c)) (k Cubical.Data.Nat.+ l Cubical.Data.Nat.+ m , ⋆ (⋆ a b) c) - ⋆Assoc {zero} {zero} {zero} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _ ∙ ·Assoc _ _ _) - ⋆Assoc {zero} {zero} {suc m} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _) - ⋆Assoc {zero} {suc l} {m} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _) - ⋆Assoc {suc k} {l} {m} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _) + ⋆Assoc {zero} {zero} {zero} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _ ∙ ·Assoc _ _ _) + ⋆Assoc {zero} {zero} {suc m} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _) + ⋆Assoc {zero} {suc l} {m} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _) + ⋆Assoc {suc k} {l} {m} a b c = ΣPathTransport→PathΣ _ _ (+-assoc _ _ _ , transportRefl _) ⋆IdR : {k : ℕ} (a : G k) → _≡_ {A = Σ[ k ∈ ℕ ] G k} (k Cubical.Data.Nat.+ 0 , ⋆ a 1r) (k , a) - ⋆IdR {zero} a = ΣPathTransport→PathΣ _ _ (refl , (transportRefl _ ∙ ·IdR _)) - ⋆IdR {suc k} a = ΣPathTransport→PathΣ _ _ ((+-zero _) , (transportRefl _)) + ⋆IdR {zero} a = ΣPathTransport→PathΣ _ _ (refl , (transportRefl _ ∙ ·IdR _)) + ⋆IdR {suc k} a = ΣPathTransport→PathΣ _ _ ((+-zero _) , (transportRefl _)) ⋆IdL : {l : ℕ} (b : G l) → _≡_ {A = Σ[ k ∈ ℕ ] G k} (l , ⋆ 1r b) (l , b) - ⋆IdL {zero} b = ΣPathTransport→PathΣ _ _ (refl , (transportRefl _ ∙ ·IdL _)) - ⋆IdL {suc l} b = ΣPathTransport→PathΣ _ _ (refl , (transportRefl _)) + ⋆IdL {zero} b = ΣPathTransport→PathΣ _ _ (refl , (transportRefl _ ∙ ·IdL _)) + ⋆IdL {suc l} b = ΣPathTransport→PathΣ _ _ (refl , (transportRefl _)) ⋆DistR+ : {k l : ℕ} (a : G k) (b c : G l) → ⋆ a (Gstr l .AbGroupStr._+_ b c) ≡ Gstr (k Cubical.Data.Nat.+ l) .AbGroupStr._+_ (⋆ a b) (⋆ a c) diff --git a/Cubical.Algebra.Group.Abelianization.AbelianizationAsCoeq.html b/Cubical.Algebra.Group.Abelianization.AbelianizationAsCoeq.html index 9d376145a1..6256d3a132 100644 --- a/Cubical.Algebra.Group.Abelianization.AbelianizationAsCoeq.html +++ b/Cubical.Algebra.Group.Abelianization.AbelianizationAsCoeq.html @@ -241,7 +241,7 @@ → (f : GroupHom G (AbGroup→Group H)) → (compGroupHom incAbAsGroupHom (inducedHom H f) ≡ f) commutativity H f = - Σ≡Prop + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (λ i x → q x i) where q : (x : fst G) @@ -254,7 +254,7 @@ → (p : compGroupHom incAbAsGroupHom g ≡ f) → (g ≡ inducedHom H f) uniqueness H f g p = - Σ≡Prop + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (λ i x → q x i) where @@ -375,7 +375,7 @@ (GroupIso→GroupHom h) (GroupIso→GroupHom (invGroupIso h))) ≡ idGroupHom leftInvGroupHom = - Σ≡Prop + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (λ i x → r x i) @@ -411,7 +411,7 @@ (HITηAsGroupHom G) (GroupIso→GroupHom (invGroupIso h)) ≡ incAbAsGroupHom isocomm = - Σ≡Prop + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (λ i x → q x i) diff --git a/Cubical.Algebra.Group.Abelianization.Properties.html b/Cubical.Algebra.Group.Abelianization.Properties.html index 6093ece2a3..41c48ab491 100644 --- a/Cubical.Algebra.Group.Abelianization.Properties.html +++ b/Cubical.Algebra.Group.Abelianization.Properties.html @@ -276,7 +276,7 @@ → (f : GroupHom G (AbGroup→Group H)) → (compGroupHom ηAsGroupHom (inducedHom H f) ≡ f) commutativity H f = - Σ≡Prop + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (λ i x → q x i) where q : (x : fst G) @@ -289,7 +289,7 @@ → (p : compGroupHom ηAsGroupHom g ≡ f) → (g ≡ inducedHom H f) uniqueness H f g p = - Σ≡Prop + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (λ i x → q x i) where diff --git a/Cubical.Algebra.Group.GroupPath.html b/Cubical.Algebra.Group.GroupPath.html index 7bf5ffb696..45902e2c9d 100644 --- a/Cubical.Algebra.Group.GroupPath.html +++ b/Cubical.Algebra.Group.GroupPath.html @@ -160,7 +160,7 @@ ∙∙ transportTransport⁻ (ua (Group≡ G H)) q where helper : transport (sym (ua (Group≡ G H))) p ≡ transport (sym (ua (Group≡ G H))) q - helper = Σ≡Prop + helper = Σ≡Prop (λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd H)) _ _) λ _ → isPropΣ (isOfHLevelPathP' 1 (isSetΠ2 λ _ _ → is-set (snd H)) _ _) @@ -191,8 +191,8 @@ where lem : transport (λ j → GroupEquiv G (GroupPath G H .fst e (~ j))) e ≡ idGroupEquiv - lem = Σ≡Prop (λ _ → isPropIsGroupHom _ _) - (Σ≡Prop (λ _ → isPropIsEquiv _) + lem = Σ≡Prop (λ _ → isPropIsGroupHom _ _) + (Σ≡Prop (λ _ → isPropIsEquiv _) (funExt λ x → (λ i → fst (fst (fst e .snd .equiv-proof (transportRefl (fst (fst e) (transportRefl x i)) i)))) ∙ retEq (fst e) x)) diff --git a/Cubical.Algebra.Group.Instances.IntMod.html b/Cubical.Algebra.Group.Instances.IntMod.html index 03321b17e2..ce8f3935c9 100644 --- a/Cubical.Algebra.Group.Instances.IntMod.html +++ b/Cubical.Algebra.Group.Instances.IntMod.html @@ -57,9 +57,9 @@ Iso.inv (fst Bool≅ℤGroup/2) (suc (suc x) , p) = ⊥.rec (¬-<-zero (predℕ-≤-predℕ (predℕ-≤-predℕ p))) Iso.rightInv (fst Bool≅ℤGroup/2) (zero , p) = - Σ≡Prop (λ _ → isProp≤) refl + Σ≡Prop (λ _ → isProp≤) refl Iso.rightInv (fst Bool≅ℤGroup/2) (suc zero , p) = - Σ≡Prop (λ _ → isProp≤) refl + Σ≡Prop (λ _ → isProp≤) refl Iso.rightInv (fst Bool≅ℤGroup/2) (suc (suc x) , p) = ⊥.rec (¬-<-zero (predℕ-≤-predℕ (predℕ-≤-predℕ p))) Iso.leftInv (fst Bool≅ℤGroup/2) false = refl @@ -81,11 +81,11 @@ ℤ→Fin-presinv : (n : ℕ) (x : ℤ) → ℤ→Fin n (- x) ≡ -ₘ ℤ→Fin n x ℤ→Fin-presinv n (pos zero) = - Σ≡Prop (λ _ → isProp≤) ((λ _ → zero) ∙ sym (cong fst help)) + Σ≡Prop (λ _ → isProp≤) ((λ _ → zero) ∙ sym (cong fst help)) where help : (-ₘ_ {n = n} 0) ≡ 0 help = GroupTheory.inv1g (ℤGroup/ (suc n)) -ℤ→Fin-presinv n (pos (suc x)) = Σ≡Prop (λ _ → isProp≤) refl +ℤ→Fin-presinv n (pos (suc x)) = Σ≡Prop (λ _ → isProp≤) refl ℤ→Fin-presinv n (negsuc x) = sym (GroupTheory.invInv (ℤGroup/ (suc n)) _) @@ -97,7 +97,7 @@ -ₘ1-id (suc n) = cong -ₘ_ (FinPathℕ ((1 mod suc (suc n)) , mod< (suc n) 1) 1 (modIndBase (suc n) 1 (n , +-comm n 2)) .snd) - ∙ Σ≡Prop (λ _ → isProp≤) + ∙ Σ≡Prop (λ _ → isProp≤) ((+inductionBase (suc n) _ (λ x _ → ((suc (suc n)) ∸ x) mod (suc (suc n))) λ _ x → x) 1 (n , (+-comm n 2))) @@ -111,7 +111,7 @@ suc-ₘ1 (suc n) y = (λ i → ((suc y mod suc (suc n)) , mod< (suc n) (suc y)) +ₘ (-ₘ1-id (suc n) i)) - ∙ Σ≡Prop (λ _ → isProp≤) + ∙ Σ≡Prop (λ _ → isProp≤) (cong (_mod (2 +ℕ n)) (cong (_+ℕ (suc n) mod (2 +ℕ n)) (mod+mod≡mod (suc (suc n)) 1 y)) @@ -168,12 +168,12 @@ +1case (pos zero) = sym (GroupStr.·IdR (snd (ℤGroup/ (suc n))) _) +1case (pos (suc y)) = cong (ℤ→Fin n) (+Comm 1 (pos (suc y))) - ∙ Σ≡Prop (λ _ → isProp≤) (mod+mod≡mod (suc n) 1 (suc y)) + ∙ Σ≡Prop (λ _ → isProp≤) (mod+mod≡mod (suc n) 1 (suc y)) +1case (negsuc zero) = - Σ≡Prop (λ _ → isProp≤) refl + Σ≡Prop (λ _ → isProp≤) refl ∙ sym (GroupStr.·InvR (snd (ℤGroup/ (suc n))) (modInd n 1 , mod< n 1)) +1case (negsuc (suc y)) = - Σ≡Prop (λ _ → isProp≤) + Σ≡Prop (λ _ → isProp≤) (cong fst (cong (ℤ→Fin n) (+Comm 1 (negsuc (suc y)))) ∙∙ cong fst (cong -ₘ_ (refl {x = suc y mod suc n , mod< n (suc y)})) ∙∙ cong fst (sym (1-ₘsuc n (suc y))) @@ -199,7 +199,7 @@ → (modInd n 1 , mod< n 1) +ₘ (modInd n (suc x) , mod< n (suc x)) ≡ ℤ→Fin n (pos (suc (suc x))) lem x = - Σ≡Prop (λ _ → isProp≤) (sym (mod+mod≡mod (suc n) 1 (suc x))) + Σ≡Prop (λ _ → isProp≤) (sym (mod+mod≡mod (suc n) 1 (suc x))) -- ℤ/2 lemmas ℤ/2-elim : ∀ {ℓ} {A : Fin 2 → Type ℓ} → A 0 → A 1 → (x : _) → A x diff --git a/Cubical.Algebra.Group.IsomorphismTheorems.html b/Cubical.Algebra.Group.IsomorphismTheorems.html index e93adf10a2..f0a9cebeca 100644 --- a/Cubical.Algebra.Group.IsomorphismTheorems.html +++ b/Cubical.Algebra.Group.IsomorphismTheorems.html @@ -74,7 +74,7 @@ f2 : ⟨ G / kerNormalSubgroup ⟩ → ⟨ imϕ ⟩ f2 = recS imG.is-set (λ y → ϕ .fst y , ∣ y , refl ∣₁) - (λ x y r → Σ≡Prop (λ _ → squash₁) + (λ x y r → Σ≡Prop (λ _ → squash₁) (rem x y r)) where rem : (x y : ⟨ G ⟩) → ϕ .fst (x G.· G.inv y) ≡ H.1g → ϕ .fst x ≡ ϕ .fst y @@ -94,7 +94,7 @@ f21 : (x : ⟨ imϕ ⟩) → f2 (f1 x) ≡ x f21 (x , hx) = elim {P = λ hx → f2 (f1 (x , hx)) ≡ (x , hx)} (λ _ → imG.is-set _ _) - (λ {(x , hx) → Σ≡Prop (λ _ → squash₁) hx}) + (λ {(x , hx) → Σ≡Prop (λ _ → squash₁) hx}) hx f1-isHom : (x y : ⟨ imϕ ⟩) → f1 (x imG.· y) ≡ f1 x kerG.· f1 y @@ -121,7 +121,7 @@ (uncurry (λ h → elim (λ _ → isPropΠ (λ _ → imG.is-set _ _)) (uncurry λ g y - → λ inker → Σ≡Prop (λ _ → squash₁) inker))) + → λ inker → Σ≡Prop (λ _ → squash₁) inker))) λ b → map (λ x → (b , ∣ x ∣₁) , refl) (surj b)) where indHom : GroupHom imϕ H diff --git a/Cubical.Algebra.Group.MorphismProperties.html b/Cubical.Algebra.Group.MorphismProperties.html index c5b86ed9db..2a23d88fd1 100644 --- a/Cubical.Algebra.Group.MorphismProperties.html +++ b/Cubical.Algebra.Group.MorphismProperties.html @@ -157,7 +157,7 @@ isInjective→isContrKer : (f : GroupHom G H) → isInjective f → isContr (Ker f) fst (isInjective→isContrKer {G = G} f hf) = 1g (snd G) , f .snd .pres1 snd (isInjective→isContrKer {G = G} f hf) k = - Σ≡Prop (isPropIsInKer f) (sym (isInjective→isMono f hf (k .snd ∙ sym (f .snd .pres1)))) + Σ≡Prop (isPropIsInKer f) (sym (isInjective→isMono f hf (k .snd ∙ sym (f .snd .pres1)))) isContrKer→isInjective : (f : GroupHom G H) → isContr (Ker f) → isInjective f isContrKer→isInjective {G = G} f ((a , b) , c) x y = cong fst (sym (c (x , y)) ∙ rem) @@ -225,7 +225,7 @@ GroupEquivDirProd : {A : Group ℓ} {B : Group ℓ'} {C : Group ℓ''} {D : Group ℓ'''} → GroupEquiv A C → GroupEquiv B D → GroupEquiv (DirProd A B) (DirProd C D) -fst (GroupEquivDirProd eq1 eq2) = ≃-× (fst eq1) (fst eq2) +fst (GroupEquivDirProd eq1 eq2) = ≃-× (fst eq1) (fst eq2) snd (GroupEquivDirProd eq1 eq2) = GroupHomDirProd (_ , eq1 .snd) (_ , eq2 .snd) .snd GroupEquiv≡ : {f g : GroupEquiv G H} → fst f ≡ fst g → f ≡ g @@ -294,7 +294,7 @@ helper : (b : _) → isProp (Σ[ a ∈ ⟨ G ⟩ ] f a ≡ b) helper _ (a , ha) (b , hb) = - Σ≡Prop (λ _ → is-set (snd H) _ _) + Σ≡Prop (λ _ → is-set (snd H) _ _) (isInjective→isMono (fun i) (inj i) (ha ∙ sym hb) ) grIso : GroupIso G H diff --git a/Cubical.Algebra.Group.QuotientGroup.html b/Cubical.Algebra.Group.QuotientGroup.html index 54c978637f..70e48fd8d6 100644 --- a/Cubical.Algebra.Group.QuotientGroup.html +++ b/Cubical.Algebra.Group.QuotientGroup.html @@ -119,7 +119,7 @@ Code g = elim (λ _ → isSetHProp) (λ a → (g ≡ a) , is-set _ _) - λ a b r → Σ≡Prop (λ _ → isPropIsProp) (cong (g ≡_) (contrH a b r)) + λ a b r → Σ≡Prop (λ _ → isPropIsProp) (cong (g ≡_) (contrH a b r)) decode : (g : G) (x : G/H') → [ g ] ≡ x → Code g x .fst decode g x = J (λ x _ → Code g x .fst) refl diff --git a/Cubical.Algebra.Group.ZAction.html b/Cubical.Algebra.Group.ZAction.html index 8bf1376c23..3dd02f6e46 100644 --- a/Cubical.Algebra.Group.ZAction.html +++ b/Cubical.Algebra.Group.ZAction.html @@ -401,15 +401,15 @@ → (e ≡ idGroupEquiv) ⊎ (e ≡ negEquivℤ) characℤ≅ℤ e = ⊎-rec - (λ p → inl (Σ≡Prop (λ _ → isPropIsGroupHom _ _) - (Σ≡Prop (λ _ → isPropIsEquiv _) + (λ p → inl (Σ≡Prop (λ _ → isPropIsGroupHom _ _) + (Σ≡Prop (λ _ → isPropIsEquiv _) (funExt λ x → cong (e .fst .fst) (·Comm 1 x) ∙ GroupHomℤ→ℤPres· (fst (fst e) , snd e) x 1 ∙ cong (x *_) p ∙ ·Comm x 1)))) - (λ p → inr (Σ≡Prop (λ _ → isPropIsGroupHom _ _) - (Σ≡Prop (λ _ → isPropIsEquiv _) + (λ p → inr (Σ≡Prop (λ _ → isPropIsGroupHom _ _) + (Σ≡Prop (λ _ → isPropIsEquiv _) (funExt λ x → cong (fst (fst e)) (sym (ℤ.·IdR x)) ∙ GroupHomℤ→ℤPres· ((fst (fst e)) , (snd e)) x 1 @@ -458,7 +458,7 @@ -- Characterisation of ℤ→ℤ characGroupHomℤ : (e : GroupHom ℤGroup ℤGroup) → e ≡ ℤHom (fst e (pos 1)) characGroupHomℤ e = - Σ≡Prop (λ _ → isPropIsGroupHom _ _) + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ { (pos n) → lem n ; (negsuc n) → presinv (snd e) (pos (suc n)) @@ -506,13 +506,13 @@ where lem : (x : ℤ) → ℤ→Fin n (pos (suc n) * x) ≡ 0 lem (pos x) = cong (ℤ→Fin n) (sym (pos· (suc n) x)) - ∙ Σ≡Prop (λ _ → isProp≤) + ∙ Σ≡Prop (λ _ → isProp≤) (cong (_mod (suc n)) (·-comm (suc n) x) ∙ zero-charac-gen (suc n) x) lem (negsuc x) = cong (ℤ→Fin n) (pos·negsuc (suc n) x ∙ cong -_ (sym (pos· (suc n) (suc x)))) - ∙∙ cong -ₘ_ (Σ≡Prop (λ _ → isProp≤) + ∙∙ cong -ₘ_ (Σ≡Prop (λ _ → isProp≤) (cong (_mod (suc n)) (·-comm (suc n) (suc x)) ∙ zero-charac-gen (suc n) (suc x))) ∙∙ GroupTheory.inv1g (ℤGroup/ (suc n)) @@ -545,7 +545,7 @@ ∙∙ cong -_ (cong pos (≡remainder+quotient (suc n) (suc x))))) ∣₁}) BijectionIso.surj (ℤHom→ℤ/im≅ℤ/im1 n p) x = ∣ [ pos (fst x) ] - , (Σ≡Prop (λ _ → isProp≤) (modIndBase n (fst x) (snd x))) ∣₁ + , (Σ≡Prop (λ _ → isProp≤) (modIndBase n (fst x) (snd x))) ∣₁ -- main result ℤ/imIso : (f : GroupHom ℤGroup ℤGroup) @@ -566,9 +566,9 @@ extendHom = compGroupHom f (fst (fst negEquivℤ) , snd negEquivℤ) lem1 : imℤHomSubGroup f ≡ imℤHomSubGroup extendHom - lem1 = Σ≡Prop (λ _ → isPropIsNormal _) - (Σ≡Prop (λ _ → isPropIsSubgroup _ _) - (funExt λ x → Σ≡Prop (λ _ → isPropIsProp) + lem1 = Σ≡Prop (λ _ → isPropIsNormal _) + (Σ≡Prop (λ _ → isPropIsSubgroup _ _) + (funExt λ x → Σ≡Prop (λ _ → isPropIsProp) (isoToPath (iso (Prop.map (λ { (x , q) → (- x) , cong -_ (presinv (snd f) x) ∙ GroupTheory.invInv ℤGroup (fst f x) @@ -587,9 +587,9 @@ private imf≡kerg : imℤHomSubGroup f ≡ kerNormalSubgroup g imf≡kerg = - Σ≡Prop (λ _ → isPropIsNormal _) - (Σ≡Prop (λ _ → isPropIsSubgroup _ _) - (funExt λ x → Σ≡Prop (λ _ → isPropIsProp) + Σ≡Prop (λ _ → isPropIsNormal _) + (Σ≡Prop (λ _ → isPropIsSubgroup _ _) + (funExt λ x → Σ≡Prop (λ _ → isPropIsProp) (isoToPath (isProp→Iso (isPropIsInIm _ _) diff --git a/Cubical.Algebra.Lattice.Properties.html b/Cubical.Algebra.Lattice.Properties.html index b01f5f5051..63d9059396 100644 --- a/Cubical.Algebra.Lattice.Properties.html +++ b/Cubical.Algebra.Lattice.Properties.html @@ -94,7 +94,7 @@ _ = snd M LatticeHom≡f : fst φ ≡ fst ψ → φ ≡ ψ - LatticeHom≡f = Σ≡Prop λ f → isPropIsLatticeHom _ f _ + LatticeHom≡f = Σ≡Prop λ f → isPropIsLatticeHom _ f _ diff --git a/Cubical.Algebra.Matrix.CommRingCoefficient.html b/Cubical.Algebra.Matrix.CommRingCoefficient.html index b357d3696b..48e175cf2a 100644 --- a/Cubical.Algebra.Matrix.CommRingCoefficient.html +++ b/Cubical.Algebra.Matrix.CommRingCoefficient.html @@ -117,7 +117,7 @@ isInv {n = n} M = Σ[ N ∈ Mat n n ] isInv' M N isPropIsInv : (M : Mat n n) → isProp (isInv M) - isPropIsInv M p q = Σ≡Prop (λ _ → isPropIsInv' M _) (invUniq M _ _ (p .snd) (q .snd)) + isPropIsInv M p q = Σ≡Prop (λ _ → isPropIsInv' M _) (invUniq M _ _ (p .snd) (q .snd)) isInv⋆ : {M M' : Mat n n} → isInv M → isInv M' → isInv (M ⋆ M') isInv⋆ (N , p) (N' , q) .fst = N' ⋆ N diff --git a/Cubical.Algebra.Monoid.Submonoid.html b/Cubical.Algebra.Monoid.Submonoid.html index 9095736f2f..8f60e3df8e 100644 --- a/Cubical.Algebra.Monoid.Submonoid.html +++ b/Cubical.Algebra.Monoid.Submonoid.html @@ -28,9 +28,9 @@ ε' = ε , ε-closed _·'_ = λ where (m , m∈) (n , n∈) → m · n , ·-closed m∈ n∈ is-setM' = isSetΣSndProp is-set (∈-isProp S) - ·Assoc' = λ x y z → Σ≡Prop (∈-isProp S) (·Assoc (fst x) (fst y) (fst z)) - ·IdR' = λ x → Σ≡Prop (∈-isProp S) (·IdR (fst x)) - ·IdL' = λ x → Σ≡Prop (∈-isProp S) (·IdL (fst x)) + ·Assoc' = λ x y z → Σ≡Prop (∈-isProp S) (·Assoc (fst x) (fst y) (fst z)) + ·IdR' = λ x → Σ≡Prop (∈-isProp S) (·IdR (fst x)) + ·IdL' = λ x → Σ≡Prop (∈-isProp S) (·IdL (fst x)) in makeMonoid ε' _·'_ is-setM' ·Assoc' ·IdR' ·IdL' module _ where diff --git a/Cubical.Algebra.OrderedCommMonoid.PropCompletion.html b/Cubical.Algebra.OrderedCommMonoid.PropCompletion.html index 4f8a253ca8..737a511c76 100644 --- a/Cubical.Algebra.OrderedCommMonoid.PropCompletion.html +++ b/Cubical.Algebra.OrderedCommMonoid.PropCompletion.html @@ -106,10 +106,10 @@ where seqPath : fst s ≡ fst l seqPath i n = - Σ≡Prop (λ _ → isPropIsProp) {u = fst s n} {v = fst l n} (pwPath n) i + Σ≡Prop (λ _ → isPropIsProp) {u = fst s n} {v = fst l n} (pwPath n) i path : s ≡ l - path = Σ≡Prop isPropUpwardClosed seqPath + path = Σ≡Prop isPropUpwardClosed seqPath pathFromImplications : (s l : M↑) → ((n : fst M) → typeAt n s → typeAt n l) diff --git a/Cubical.Algebra.OrderedCommMonoid.Properties.html b/Cubical.Algebra.OrderedCommMonoid.Properties.html index 84715c692c..9de010d27e 100644 --- a/Cubical.Algebra.OrderedCommMonoid.Properties.html +++ b/Cubical.Algebra.OrderedCommMonoid.Properties.html @@ -49,7 +49,7 @@ (λ x → is-refl (fst x)) (λ x y z → is-trans (fst x) (fst y) (fst z)) (λ x y x≤y y≤x - → Σ≡Prop (λ x → snd (P x)) + → Σ≡Prop (λ x → snd (P x)) (is-antisym (fst x) (fst y) (pres≤ x y x≤y) (pres≤ y x y≤x))) (λ x y z x≤y → MonotoneR (pres≤ x y x≤y)) λ x y z x≤y → MonotoneL (pres≤ x y x≤y) diff --git a/Cubical.Algebra.Polynomials.UnivariateList.Base.html b/Cubical.Algebra.Polynomials.UnivariateList.Base.html index 7964387c93..7f678e0b1b 100644 --- a/Cubical.Algebra.Polynomials.UnivariateList.Base.html +++ b/Cubical.Algebra.Polynomials.UnivariateList.Base.html @@ -165,7 +165,7 @@ shiftPolyFunPrepends0 : (p : Poly R') → shiftPolyFun (Poly→PolyFun p) ≡ Poly→PolyFun (0r ∷ p) shiftPolyFunPrepends0 p = - Σ≡Prop (λ _ → isPropPropTrunc) + Σ≡Prop (λ _ → isPropPropTrunc) λ {i zero → 0r; i (suc n) → fst (Poly→PolyFun p) n} ---------------------------------------------------- diff --git a/Cubical.Algebra.Polynomials.UnivariateList.Properties.html b/Cubical.Algebra.Polynomials.UnivariateList.Properties.html index ca84c0131a..27a37ef97c 100644 --- a/Cubical.Algebra.Polynomials.UnivariateList.Properties.html +++ b/Cubical.Algebra.Polynomials.UnivariateList.Properties.html @@ -661,7 +661,7 @@ p≡0 = p ≡⟨ sym (PolyFun→Poly→PolyFun p) ⟩ PolyFun→Poly (Poly→PolyFun p) ≡⟨ cong PolyFun→Poly - (Poly→PolyFun p ≡⟨ ( Σ≡Prop (λ (f : ℕ → R) → isPropPropTrunc) λ i n → Funp≡0 n i) ⟩ + (Poly→PolyFun p ≡⟨ ( Σ≡Prop (λ (f : ℕ → R) → isPropPropTrunc) λ i n → Funp≡0 n i) ⟩ Poly→PolyFun [] ∎) ⟩ PolyFun→Poly (Poly→PolyFun []) ≡⟨ PolyFun→Poly→PolyFun 0P ⟩ 0P ∎ diff --git a/Cubical.Algebra.Ring.Base.html b/Cubical.Algebra.Ring.Base.html index ff1980de63..7754940f44 100644 --- a/Cubical.Algebra.Ring.Base.html +++ b/Cubical.Algebra.Ring.Base.html @@ -205,7 +205,7 @@ RingHomPathP R S T p φ ψ q = ΣPathP (q , isProp→PathP (λ _ → isPropIsRingHom _ _ _) _ _) RingHom≡ : {R : Ring ℓ} {S : Ring ℓ'} {φ ψ : RingHom R S} → fst φ ≡ fst ψ → φ ≡ ψ -RingHom≡ = Σ≡Prop λ f → isPropIsRingHom _ f _ +RingHom≡ = Σ≡Prop λ f → isPropIsRingHom _ f _ 𝒮ᴰ-Ring : DUARel (𝒮-Univ ℓ) RingStr ℓ 𝒮ᴰ-Ring = diff --git a/Cubical.Algebra.Ring.DirectProd.html b/Cubical.Algebra.Ring.DirectProd.html index 832bb05d01..9081b6a08e 100644 --- a/Cubical.Algebra.Ring.DirectProd.html +++ b/Cubical.Algebra.Ring.DirectProd.html @@ -101,7 +101,7 @@ _·Y×Y'_ = _·_ (snd (DirectProd-Ring Yr Y'r)) re×re' : (re : RingEquiv Xr Yr) → (re' : RingEquiv X'r Y'r) → (X × X') ≃ (Y × Y') - re×re' re re' = ≃-× (fst re) (fst re') + re×re' re re' = ≃-× (fst re) (fst re') Coproduct-Equiv-12 : (re : RingEquiv Xr Yr) → (re' : RingEquiv X'r Y'r) → RingEquiv (DirectProd-Ring Xr X'r) (DirectProd-Ring Yr Y'r) diff --git a/Cubical.Algebra.Ring.Properties.html b/Cubical.Algebra.Ring.Properties.html index 48f7ffa89e..fb9aa8d5dc 100644 --- a/Cubical.Algebra.Ring.Properties.html +++ b/Cubical.Algebra.Ring.Properties.html @@ -296,7 +296,7 @@ ∙∙ transportTransport⁻ (ua (Ring≡ A B)) q where helper : transport (sym (ua (Ring≡ A B))) p ≡ transport (sym (ua (Ring≡ A B))) q - helper = Σ≡Prop + helper = Σ≡Prop (λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) diff --git a/Cubical.Algebra.Ring.Quotient.html b/Cubical.Algebra.Ring.Quotient.html index 66ba031073..14fa1ea11b 100644 --- a/Cubical.Algebra.Ring.Quotient.html +++ b/Cubical.Algebra.Ring.Quotient.html @@ -9,7 +9,7 @@ open import Cubical.Foundations.Powerset using (_∈_; _⊆_; ⊆-extensionality) -- \in, \sub= open import Cubical.Functions.Surjection -open import Cubical.Data.Sigma using (Σ≡Prop) +open import Cubical.Data.Sigma using (Σ≡Prop) open import Cubical.Relation.Binary @@ -320,6 +320,6 @@ kernel≡I : {R : Ring ℓ} (I : IdealsIn R) → kernelIdeal (quotientHom R I) ≡ I -kernel≡I {R = R} I = Σ≡Prop (isPropIsIdeal R) (⊆-extensionality _ _ (ker⊆I , I⊆ker)) +kernel≡I {R = R} I = Σ≡Prop (isPropIsIdeal R) (⊆-extensionality _ _ (ker⊆I , I⊆ker)) where open idealIsKernel I \ No newline at end of file diff --git a/Cubical.Algebra.ZariskiLattice.StructureSheaf.html b/Cubical.Algebra.ZariskiLattice.StructureSheaf.html index 8adec51727..51b2173a62 100644 --- a/Cubical.Algebra.ZariskiLattice.StructureSheaf.html +++ b/Cubical.Algebra.ZariskiLattice.StructureSheaf.html @@ -150,7 +150,7 @@ ZariskiCat = DistLatticeCategory ZariskiLattice BOCat : Category ℓ ℓ - BOCat = ΣPropCat ZariskiCat BasicOpens + BOCat = ΣPropCat ZariskiCat BasicOpens private P : ZL → Type _ @@ -168,42 +168,42 @@ open PreSheafFromUniversalProp ZariskiCat P 𝓕 uniqueHom - 𝓞ᴮ : Functor (BOCat ^op) CommRingsCategory - 𝓞ᴮ = funcComp (ForgetfulCommAlgebra→CommRing R') universalPShf + 𝓞ᴮ : Functor (BOCat ^op) CommRingsCategory + 𝓞ᴮ = funcComp (ForgetfulCommAlgebra→CommRing R') universalPShf -- The extension - open Functor + open Functor open PreSheafExtension ZariskiLattice CommRingsCategory LimitsCommRingsCategory BasicOpens - 𝓞 : Functor (ZariskiCat ^op) CommRingsCategory + 𝓞 : Functor (ZariskiCat ^op) CommRingsCategory 𝓞 = DLRan 𝓞ᴮ - toBasisPath : ∀ f → 𝓞 .F-ob (D f) ≡ 𝓞ᴮ .F-ob (D f , ∣ f , refl ∣₁) - toBasisPath f = cong (λ F → F .F-ob (D f , ∣ f , refl ∣₁)) - (NatIsoToPath isUnivalentCommRingsCategory (DLRanNatIso 𝓞ᴮ)) + toBasisPath : ∀ f → 𝓞 .F-ob (D f) ≡ 𝓞ᴮ .F-ob (D f , ∣ f , refl ∣₁) + toBasisPath f = cong (λ F → F .F-ob (D f , ∣ f , refl ∣₁)) + (NatIsoToPath isUnivalentCommRingsCategory (DLRanNatIso 𝓞ᴮ)) open InvertingElementsBase R' private Forgetful = ForgetfulCommAlgebra→CommRing R' {ℓ' = ℓ} - 𝓞ᴮOb≡ : ∀ f → 𝓞ᴮ .F-ob (D f , ∣ f , refl ∣₁) ≡ R[1/ f ]AsCommRing - 𝓞ᴮOb≡ f = 𝓞ᴮ .F-ob (D f , ∣ f , refl ∣₁) ≡⟨ refl ⟩ + 𝓞ᴮOb≡ : ∀ f → 𝓞ᴮ .F-ob (D f , ∣ f , refl ∣₁) ≡ R[1/ f ]AsCommRing + 𝓞ᴮOb≡ f = 𝓞ᴮ .F-ob (D f , ∣ f , refl ∣₁) ≡⟨ refl ⟩ -- all of this should hold by refl ----------------------------------------------------------- -- but somehow Agda takes forever to type-check if you don't use ----------------------------- -- the lemma funcCompOb≡ (which is just refl itself) or if you leave out --------------------- -- any of the intermediate refl steps -------------------------------------------------------- - (funcComp (ForgetfulCommAlgebra→CommRing R') universalPShf) .F-ob (D f , ∣ f , refl ∣₁) - ≡⟨ funcCompOb≡ Forgetful universalPShf _ ⟩ - Forgetful .F-ob R[1/ f ]AsCommAlgebra + (funcComp (ForgetfulCommAlgebra→CommRing R') universalPShf) .F-ob (D f , ∣ f , refl ∣₁) + ≡⟨ funcCompOb≡ Forgetful universalPShf _ ⟩ + Forgetful .F-ob R[1/ f ]AsCommAlgebra ≡⟨ refl ⟩ ---------------------------------------------------------------------------------------------- CommAlgebra→CommRing R[1/ f ]AsCommAlgebra ≡⟨ invElCommAlgebra→CommRingPath f ⟩ R[1/ f ]AsCommRing ∎ - baseSections : ∀ f → 𝓞 .F-ob (D f) ≡ R[1/ f ]AsCommRing + baseSections : ∀ f → 𝓞 .F-ob (D f) ≡ R[1/ f ]AsCommRing baseSections f = toBasisPath f ∙ 𝓞ᴮOb≡ f - globalSection : 𝓞 .F-ob (D 1r) ≡ R' + globalSection : 𝓞 .F-ob (D 1r) ≡ R' globalSection = baseSections 1r ∙ invertingUnitsPath _ _ (Units.RˣContainsOne _) @@ -219,15 +219,15 @@ λ φ _ → isTerminal𝓞ᴮ[0] A .snd φ where -- D(0) is not 0 of the Zariski lattice by refl! - p : 𝓞ᴮ .F-ob (0l , isBO⋁α) ≡ R[1/ 0r ]AsCommRing - p = 𝓞ᴮ .F-ob (0l , isBO⋁α) - ≡⟨ cong (𝓞ᴮ .F-ob) (Σ≡Prop (λ _ → ∈ₚ-isProp _ _) + p : 𝓞ᴮ .F-ob (0l , isBO⋁α) ≡ R[1/ 0r ]AsCommRing + p = 𝓞ᴮ .F-ob (0l , isBO⋁α) + ≡⟨ cong (𝓞ᴮ .F-ob) (Σ≡Prop (λ _ → ∈ₚ-isProp _ _) (eq/ _ _ ((λ ()) , λ {zero → ∣ 1 , ∣ (λ ()) , 0LeftAnnihilates _ ∣₁ ∣₁ }))) ⟩ - 𝓞ᴮ .F-ob (D 0r , ∣ 0r , refl ∣₁) + 𝓞ᴮ .F-ob (D 0r , ∣ 0r , refl ∣₁) ≡⟨ 𝓞ᴮOb≡ 0r ⟩ R[1/ 0r ]AsCommRing ∎ - isTerminal𝓞ᴮ[0] : isTerminal CommRingsCategory (𝓞ᴮ .F-ob (0l , isBO⋁α)) + isTerminal𝓞ᴮ[0] : isTerminal CommRingsCategory (𝓞ᴮ .F-ob (0l , isBO⋁α)) isTerminal𝓞ᴮ[0] = subst (isTerminal CommRingsCategory) (sym (p ∙ R[1/0]≡0)) (TerminalCommRing .snd) @@ -362,25 +362,25 @@ pres+ (snd (coneOut /1/1Cone (pair i j i<j))) x y = cong [_] (≡-× (cong [_] (≡-× (cong₂ _+_ (useSolver x) (useSolver y)) - (Σ≡Prop (λ _ → isPropPropTrunc) (useSolver 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (useSolver 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) where useSolver : ∀ a → a ≡ a · 1r · (1r · 1r) useSolver = solve R' pres· (snd (coneOut /1/1Cone (pair i j i<j))) x y = cong [_] (≡-× (cong [_] (≡-× refl - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r))))) - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) pres- (snd (coneOut /1/1Cone (pair i j i<j))) _ = refl coneOutCommutes /1/1Cone idAr = idCompCommRingHom _ coneOutCommutes /1/1Cone singPairL = RingHom≡ (funExt (λ x → cong [_] (≡-× (cong [_] (≡-× (cong (x ·_) (transportRefl 1r) ∙ ·IdR x) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) coneOutCommutes /1/1Cone singPairR = RingHom≡ (funExt (λ x → cong [_] (≡-× (cong [_] (≡-× (cong (x ·_) (transportRefl 1r) ∙ ·IdR x) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) open LimitFromCommRing R' R[1/ h ]AsCommRing (DLShfDiag (suc n) ℓ) doubleLocDiag doubleLocCone /1/1Cone @@ -398,7 +398,7 @@ -- we only give the paths on objects -- R[1/h][1/fᵢ] ≡ [1/fᵢ] -- R[1/h][1/fᵢfⱼ] ≡ R[1/fᵢfⱼ] - algPaths : ∀ v → F-ob algDiag v ≡ F-ob (funcComp universalPShf BDiag) v + algPaths : ∀ v → F-ob algDiag v ≡ F-ob (funcComp universalPShf BDiag) v algPaths (sing i) = doubleLocCancel (f∈√⟨h⟩ i) where open DoubleAlgLoc R' h (f i) @@ -411,7 +411,7 @@ R[1/h][1/fᵢfⱼ]AsCommRingReg = InvertingElementsBase.R[1/_]AsCommRing R[1/ h ]AsCommRing ((f i · f j) /1) - path : toCommAlg ( F-ob doubleLocDiag (pair i j i<j) + path : toCommAlg ( F-ob doubleLocDiag (pair i j i<j) , coneOut /1/1Cone (pair i j i<j)) ≡ toCommAlg (R[1/h][1/fᵢfⱼ]AsCommRingReg , /1/1AsCommRingHom (f i · f j)) path = cong toCommAlg (ΣPathP (p , q)) @@ -419,15 +419,15 @@ eqInR[1/h] : (f i /1) · (f j /1) ≡ (f i · f j) /1 eqInR[1/h] = sym (/1AsCommRingHom .snd .pres· (f i) (f j)) - p : F-ob doubleLocDiag (pair i j i<j) ≡ R[1/h][1/fᵢfⱼ]AsCommRingReg + p : F-ob doubleLocDiag (pair i j i<j) ≡ R[1/h][1/fᵢfⱼ]AsCommRingReg p i = InvertingElementsBase.R[1/_]AsCommRing R[1/ h ]AsCommRing (eqInR[1/h] i) q : PathP (λ i → CommRingHom R' (p i)) (coneOut /1/1Cone (pair i j i<j)) (/1/1AsCommRingHom (f i · f j)) q = toPathP (RingHom≡ (funExt ( λ r → cong [_] (≡-× (cong [_] (≡-× (transportRefl _ ∙ transportRefl r) - (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))))) -- our main result diff --git a/Cubical.Algebra.ZariskiLattice.StructureSheafPullback.html b/Cubical.Algebra.ZariskiLattice.StructureSheafPullback.html index 9e8aef953c..675a4df8f1 100644 --- a/Cubical.Algebra.ZariskiLattice.StructureSheafPullback.html +++ b/Cubical.Algebra.ZariskiLattice.StructureSheafPullback.html @@ -133,7 +133,7 @@ ZariskiCat = DistLatticeCategory ZariskiLattice BOCat : Category ℓ ℓ - BOCat = ΣPropCat ZariskiCat BasicOpens + BOCat = ΣPropCat ZariskiCat BasicOpens private P : ZL → Type _ @@ -171,7 +171,7 @@ open PreSheafFromUniversalProp ZariskiCat P 𝓕 uniqueHom - BasisStructurePShf : Functor (BOCat ^op) (CommAlgebrasCategory R') + BasisStructurePShf : Functor (BOCat ^op) (CommAlgebrasCategory R') BasisStructurePShf = universalPShf @@ -182,10 +182,10 @@ -- only proof for weak notion of sheaf on a basis isSheafBasisStructurePShf : isDLBasisSheafPullback BasisStructurePShf fst isSheafBasisStructurePShf 0∈BO = subst (isTerminal (CommAlgebrasCategory R')) - (sym R[1/0]≡0 ∙ λ i → F-ob (0z , canonical0∈BO≡0∈BO i)) + (sym R[1/0]≡0 ∙ λ i → F-ob (0z , canonical0∈BO≡0∈BO i)) (TerminalCommAlgebra R' .snd) where - open Functor ⦃...⦄ + open Functor ⦃...⦄ instance _ = BasisStructurePShf @@ -237,15 +237,15 @@ -- write everything explicitly so things can type-check thePShfCospan : (a : Σ[ f ∈ R ] D f ≡ 𝔞) (b : Σ[ g ∈ R ] D g ≡ 𝔟) → Cospan (CommAlgebrasCategory R') - Cospan.l (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-ob (𝔟 , ∣ g , Dg≡𝔟 ∣₁) - Cospan.m (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-ob + Cospan.l (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-ob (𝔟 , ∣ g , Dg≡𝔟 ∣₁) + Cospan.m (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-ob (𝔞 ∧z 𝔟 , basicOpensAreBasis .∧lClosed 𝔞 𝔟 ∣ f , Df≡𝔞 ∣₁ ∣ g , Dg≡𝔟 ∣₁) - Cospan.r (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-ob (𝔞 , ∣ f , Df≡𝔞 ∣₁) - Cospan.s₁ (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-hom + Cospan.r (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-ob (𝔞 , ∣ f , Df≡𝔞 ∣₁) + Cospan.s₁ (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-hom {x = (𝔟 , ∣ g , Dg≡𝔟 ∣₁)} {y = (𝔞 ∧z 𝔟 , basicOpensAreBasis .∧lClosed 𝔞 𝔟 ∣ f , Df≡𝔞 ∣₁ ∣ g , Dg≡𝔟 ∣₁)} (hom-∧₂ ZariskiLattice (CommAlgebrasCategory R' {ℓ' = ℓ}) 𝔞 𝔟) - Cospan.s₂ (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-hom + Cospan.s₂ (thePShfCospan (f , Df≡𝔞) (g , Dg≡𝔟)) = BasisStructurePShf .Functor.F-hom {x = (𝔞 , ∣ f , Df≡𝔞 ∣₁)} {y = (𝔞 ∧z 𝔟 , basicOpensAreBasis .∧lClosed 𝔞 𝔟 ∣ f , Df≡𝔞 ∣₁ ∣ g , Dg≡𝔟 ∣₁)} (hom-∧₁ ZariskiLattice (CommAlgebrasCategory R' {ℓ' = ℓ}) 𝔞 𝔟) @@ -373,14 +373,14 @@ pres1 (snd /1/1AsCommRingHomFG) = refl pres+ (snd /1/1AsCommRingHomFG) x y = cong [_] (≡-× (cong [_] (≡-× (cong₂ _+_ (useSolver x) (useSolver y)) - (Σ≡Prop (λ _ → isPropPropTrunc) (useSolver 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (useSolver 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) where useSolver : ∀ a → a ≡ a · 1r · (1r · 1r) useSolver = solve R' pres· (snd /1/1AsCommRingHomFG) x y = cong [_] (≡-× (cong [_] (≡-× refl - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r))))) - (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (sym (·IdR 1r)))) pres- (snd /1/1AsCommRingHomFG) x = refl open Cospan @@ -395,14 +395,14 @@ isRHomR[1/h][1/f]→R[1/h][1/fg] : theRingCospan .s₂ ∘r /1/1AsCommRingHom f ≡ /1/1AsCommRingHomFG isRHomR[1/h][1/f]→R[1/h][1/fg] = RingHom≡ (funExt (λ x → cong [_] (≡-× (cong [_] (≡-× (cong (x ·_) (transportRefl 1r) ∙ ·IdR x) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) isRHomR[1/h][1/g]→R[1/h][1/fg] : theRingCospan .s₁ ∘r /1/1AsCommRingHom g ≡ /1/1AsCommRingHomFG isRHomR[1/h][1/g]→R[1/h][1/fg] = RingHom≡ (funExt (λ x → cong [_] (≡-× (cong [_] (≡-× (cong (x ·_) (transportRefl 1r) ∙ ·IdR x) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (cong (1r ·_) (transportRefl 1r) ∙ ·IdR 1r))))) open PullbackFromCommRing R' theRingCospan theRingPullback @@ -448,6 +448,6 @@ q : PathP (λ i → CommRingHom R' (p i)) /1/1AsCommRingHomFG (/1/1AsCommRingHom (f · g)) q = toPathP (RingHom≡ (funExt ( λ x → cong [_] (≡-× (cong [_] (≡-× (transportRefl _ ∙ transportRefl x) - (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))) - (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))))) + (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))) + (Σ≡Prop (λ _ → isPropPropTrunc) (transportRefl 1r)))))) \ No newline at end of file diff --git a/Cubical.Algebra.ZariskiLattice.UniversalProperty.html b/Cubical.Algebra.ZariskiLattice.UniversalProperty.html index 301015d1fa..8e49bdad2e 100644 --- a/Cubical.Algebra.ZariskiLattice.UniversalProperty.html +++ b/Cubical.Algebra.ZariskiLattice.UniversalProperty.html @@ -277,7 +277,7 @@ χunique : (y : Σ[ χ' ∈ DistLatticeHom ZariskiLattice L' ] fst χ' ∘ D ≡ d) → (χ , funExt χcomp) ≡ y - χunique (χ' , χ'∘D≡d) = Σ≡Prop (λ _ → isSetΠ (λ _ → isSetL) _ _) (LatticeHom≡f _ _ + χunique (χ' , χ'∘D≡d) = Σ≡Prop (λ _ → isSetΠ (λ _ → isSetL) _ _) (LatticeHom≡f _ _ (funExt (elimProp (λ _ → isSetL _ _) (uncurry uniqHelper)))) where uniqHelper : (n : ℕ) (α : FinVec R n) → fst χ [ n , α ] ≡ fst χ' [ n , α ] diff --git a/Cubical.Categories.Adjoint.html b/Cubical.Categories.Adjoint.html index 8aeb18d0e9..079c7dc937 100644 --- a/Cubical.Categories.Adjoint.html +++ b/Cubical.Categories.Adjoint.html @@ -4,19 +4,22 @@ module Cubical.Categories.Adjoint where open import Cubical.Foundations.Prelude -open import Cubical.Data.Sigma -open import Cubical.Categories.Category -open import Cubical.Categories.Functor -open import Cubical.Categories.NaturalTransformation -open import Cubical.Foundations.Isomorphism -open import Cubical.Foundations.Univalence +open import Cubical.Foundations.Equiv -open Functor +open import Cubical.Data.Sigma +open import Cubical.Categories.Category +open import Cubical.Categories.Functor +open import Cubical.Categories.Instances.Functors +open import Cubical.Categories.NaturalTransformation +open import Cubical.Foundations.Isomorphism +open import Cubical.Foundations.Univalence -open Iso -open Category +open Functor -{- +open Iso +open Category + +{- ============================================== Overview ============================================== @@ -27,11 +30,11 @@ equivalence. -} -private - variable - ℓC ℓC' ℓD ℓD' : Level +private + variable + ℓC ℓC' ℓD ℓD' : Level -{- +{- ============================================== Adjoint definitions ============================================== @@ -42,76 +45,195 @@ definition. -} -module UnitCounit {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) (G : Functor D C) where - record TriangleIdentities - (η : 𝟙⟨ C ⟩ ⇒ (funcComp G F)) - (ε : (funcComp F G) ⇒ 𝟙⟨ D ⟩) - : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) - where - field - Δ₁ : ∀ c → F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε ⟦ F ⟅ c ⟆ ⟧ ≡ D .id - Δ₂ : ∀ d → η ⟦ G ⟅ d ⟆ ⟧ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫ ≡ C .id - - -- Adjoint def 1: unit-counit - record _⊣_ : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where - field - -- unit - η : 𝟙⟨ C ⟩ ⇒ (funcComp G F) - -- counit - ε : (funcComp F G) ⇒ 𝟙⟨ D ⟩ - triangleIdentities : TriangleIdentities η ε - open TriangleIdentities triangleIdentities public - -module NaturalBijection where - -- Adjoint def 2: natural bijection - record _⊣_ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) (G : Functor D C) : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where - field - adjIso : ∀ {c d} → Iso (D [ F ⟅ c ⟆ , d ]) (C [ c , G ⟅ d ⟆ ]) - - infix 40 _♭ - infix 40 _♯ - _♭ : ∀ {c d} → (D [ F ⟅ c ⟆ , d ]) → (C [ c , G ⟅ d ⟆ ]) - (_♭) {_} {_} = adjIso .fun - - _♯ : ∀ {c d} → (C [ c , G ⟅ d ⟆ ]) → (D [ F ⟅ c ⟆ , d ]) - (_♯) {_} {_} = adjIso .inv - - field - adjNatInD : ∀ {c : C .ob} {d d'} (f : D [ F ⟅ c ⟆ , d ]) (k : D [ d , d' ]) - → (f ⋆⟨ D ⟩ k) ♭ ≡ f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ - - adjNatInC : ∀ {c' c d} (g : C [ c , G ⟅ d ⟆ ]) (h : C [ c' , c ]) - → (h ⋆⟨ C ⟩ g) ♯ ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯ - - adjNatInD' : ∀ {c : C .ob} {d d'} (g : C [ c , G ⟅ d ⟆ ]) (k : D [ d , d' ]) - → g ♯ ⋆⟨ D ⟩ k ≡ (g ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ - adjNatInD' {c} {d} {d'} g k = - g ♯ ⋆⟨ D ⟩ k - ≡⟨ sym (adjIso .leftInv (g ♯ ⋆⟨ D ⟩ k)) ⟩ - ((g ♯ ⋆⟨ D ⟩ k) ♭) ♯ - ≡⟨ cong _♯ (adjNatInD (g ♯) k) ⟩ - ((g ♯) ♭ ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ - ≡⟨ cong _♯ (cong (λ g' → seq' C g' (G ⟪ k ⟫)) (adjIso .rightInv g)) ⟩ - (g ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ ∎ - - adjNatInC' : ∀ {c' c d} (f : D [ F ⟅ c ⟆ , d ]) (h : C [ c' , c ]) - → h ⋆⟨ C ⟩ (f ♭) ≡ (F ⟪ h ⟫ ⋆⟨ D ⟩ f) ♭ - adjNatInC' {c'} {c} {d} f h = - h ⋆⟨ C ⟩ (f ♭) - ≡⟨ sym (adjIso .rightInv (h ⋆⟨ C ⟩ (f ♭))) ⟩ - ((h ⋆⟨ C ⟩ (f ♭)) ♯) ♭ - ≡⟨ cong _♭ (adjNatInC (f ♭) h) ⟩ - ((F ⟪ h ⟫ ⋆⟨ D ⟩ (f ♭) ♯) ♭) - ≡⟨ cong _♭ (cong (λ f' → seq' D (F ⟪ h ⟫) f') (adjIso .leftInv f)) ⟩ - (F ⟪ h ⟫ ⋆⟨ D ⟩ f) ♭ ∎ - - isLeftAdjoint : {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) - isLeftAdjoint {C = C}{D} F = Σ[ G ∈ Functor D C ] F ⊣ G - - isRightAdjoint : {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (G : Functor D C) → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) - isRightAdjoint {C = C}{D} G = Σ[ F ∈ Functor C D ] F ⊣ G - -{- +module UnitCounit {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) (G : Functor D C) where + record TriangleIdentities + (η : 𝟙⟨ C ⟩ ⇒ (funcComp G F)) + (ε : (funcComp F G) ⇒ 𝟙⟨ D ⟩) + : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) + where + field + Δ₁ : ∀ c → F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε ⟦ F ⟅ c ⟆ ⟧ ≡ D .id + Δ₂ : ∀ d → η ⟦ G ⟅ d ⟆ ⟧ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫ ≡ C .id + + -- Adjoint def 1: unit-counit + record _⊣_ : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where + field + -- unit + η : 𝟙⟨ C ⟩ ⇒ (funcComp G F) + -- counit + ε : (funcComp F G) ⇒ 𝟙⟨ D ⟩ + triangleIdentities : TriangleIdentities η ε + open TriangleIdentities triangleIdentities public + + +private + variable + C : Category ℓC ℓC' + D : Category ℓC ℓC' + + +module _ {F : Functor C D} {G : Functor D C} where + open UnitCounit + open _⊣_ + open NatTrans + open TriangleIdentities + opositeAdjunction : (F ⊣ G) → ((G ^opF) ⊣ (F ^opF)) + N-ob (η (opositeAdjunction x)) = N-ob (ε x) + N-hom (η (opositeAdjunction x)) f = sym (N-hom (ε x) f) + N-ob (ε (opositeAdjunction x)) = N-ob (η x) + N-hom (ε (opositeAdjunction x)) f = sym (N-hom (η x) f) + Δ₁ (triangleIdentities (opositeAdjunction x)) = + Δ₂ (triangleIdentities x) + Δ₂ (triangleIdentities (opositeAdjunction x)) = + Δ₁ (triangleIdentities x) + + Iso⊣^opF : Iso (F ⊣ G) ((G ^opF) ⊣ (F ^opF)) + fun Iso⊣^opF = opositeAdjunction + inv Iso⊣^opF = _ + rightInv Iso⊣^opF _ = refl + leftInv Iso⊣^opF _ = refl + +private + variable + F F' : Functor C D + G G' : Functor D C + + +module AdjointUniqeUpToNatIso where + open UnitCounit + module Left + (F⊣G : _⊣_ {D = D} F G) + (F'⊣G : F' ⊣ G) where + open NatTrans + + private + variable + H H' : Functor C D + + _D⋆_ = seq' D + + m : (H⊣G : H ⊣ G) (H'⊣G : H' ⊣ G) → + ∀ {x} → D [ H ⟅ x ⟆ , H' ⟅ x ⟆ ] + m {H = H} H⊣G H'⊣G = + H ⟪ (H'⊣G .η) ⟦ _ ⟧ ⟫ ⋆⟨ D ⟩ (H⊣G .ε) ⟦ _ ⟧ where open _⊣_ + + private + s : (H⊣G : H ⊣ G) (H'⊣G : H' ⊣ G) → ∀ {x} → + seq' D (m H'⊣G H⊣G {x}) (m H⊣G H'⊣G {x}) + ≡ D .id + s {H = H} {H' = H'} H⊣G H'⊣G = by-N-homs ∙ by-Δs + where + open _⊣_ H⊣G using (η ; Δ₂) + open _⊣_ H'⊣G using (ε ; Δ₁) + by-N-homs = + AssocCong₂⋆R {C = D} _ + (AssocCong₂⋆L {C = D} (sym (N-hom ε _)) _) + ∙ cong₂ _D⋆_ + (sym (F-seq H' _ _) + ∙∙ cong (H' ⟪_⟫) ((sym (N-hom η _))) + ∙∙ F-seq H' _ _) + (sym (N-hom ε _)) + + by-Δs = + ⋆Assoc D _ _ _ + ∙∙ cong (H' ⟪ _ ⟫ D⋆_) + (sym (⋆Assoc D _ _ _) + ∙ cong (_D⋆ ε ⟦ _ ⟧) + ( sym (F-seq H' _ _) + ∙∙ cong (H' ⟪_⟫) (Δ₂ (H' ⟅ _ ⟆)) + ∙∙ F-id H') + ∙ ⋆IdL D _) + ∙∙ Δ₁ _ + + open NatIso + open isIso + + F≅ᶜF' : F ≅ᶜ F' + N-ob (trans F≅ᶜF') _ = _ + N-hom (trans F≅ᶜF') _ = + sym (⋆Assoc D _ _ _) + ∙∙ cong (_D⋆ (F⊣G .ε) ⟦ _ ⟧) + (sym (F-seq F _ _) + ∙∙ cong (F ⟪_⟫) (N-hom (F'⊣G .η) _) + ∙∙ (F-seq F _ _)) + ∙∙ AssocCong₂⋆R {C = D} _ (N-hom (F⊣G .ε) _) + where open _⊣_ + inv (nIso F≅ᶜF' _) = _ + sec (nIso F≅ᶜF' _) = s F⊣G F'⊣G + ret (nIso F≅ᶜF' _) = s F'⊣G F⊣G + + F≡F' : isUnivalent D → F ≡ F' + F≡F' univD = + isUnivalent.CatIsoToPath + (isUnivalentFUNCTOR _ _ univD) + (NatIso→FUNCTORIso _ _ F≅ᶜF') + + module Right (F⊣G : F UnitCounit.⊣ G) + (F⊣G' : F UnitCounit.⊣ G') where + + G≅ᶜG' : G ≅ᶜ G' + G≅ᶜG' = Iso.inv congNatIso^opFiso + (Left.F≅ᶜF' (opositeAdjunction F⊣G') + (opositeAdjunction F⊣G)) + + open NatIso + + G≡G' : isUnivalent _ → G ≡ G' + G≡G' univC = + isUnivalent.CatIsoToPath + (isUnivalentFUNCTOR _ _ univC) + (NatIso→FUNCTORIso _ _ G≅ᶜG') + +module NaturalBijection where + -- Adjoint def 2: natural bijection + record _⊣_ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) (G : Functor D C) : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where + field + adjIso : ∀ {c d} → Iso (D [ F ⟅ c ⟆ , d ]) (C [ c , G ⟅ d ⟆ ]) + + infix 40 _♭ + infix 40 _♯ + _♭ : ∀ {c d} → (D [ F ⟅ c ⟆ , d ]) → (C [ c , G ⟅ d ⟆ ]) + (_♭) {_} {_} = adjIso .fun + + _♯ : ∀ {c d} → (C [ c , G ⟅ d ⟆ ]) → (D [ F ⟅ c ⟆ , d ]) + (_♯) {_} {_} = adjIso .inv + + field + adjNatInD : ∀ {c : C .ob} {d d'} (f : D [ F ⟅ c ⟆ , d ]) (k : D [ d , d' ]) + → (f ⋆⟨ D ⟩ k) ♭ ≡ f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ + + adjNatInC : ∀ {c' c d} (g : C [ c , G ⟅ d ⟆ ]) (h : C [ c' , c ]) + → (h ⋆⟨ C ⟩ g) ♯ ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯ + + adjNatInD' : ∀ {c : C .ob} {d d'} (g : C [ c , G ⟅ d ⟆ ]) (k : D [ d , d' ]) + → g ♯ ⋆⟨ D ⟩ k ≡ (g ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ + adjNatInD' {c} {d} {d'} g k = + g ♯ ⋆⟨ D ⟩ k + ≡⟨ sym (adjIso .leftInv (g ♯ ⋆⟨ D ⟩ k)) ⟩ + ((g ♯ ⋆⟨ D ⟩ k) ♭) ♯ + ≡⟨ cong _♯ (adjNatInD (g ♯) k) ⟩ + ((g ♯) ♭ ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ + ≡⟨ cong _♯ (cong (λ g' → seq' C g' (G ⟪ k ⟫)) (adjIso .rightInv g)) ⟩ + (g ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ ∎ + + adjNatInC' : ∀ {c' c d} (f : D [ F ⟅ c ⟆ , d ]) (h : C [ c' , c ]) + → h ⋆⟨ C ⟩ (f ♭) ≡ (F ⟪ h ⟫ ⋆⟨ D ⟩ f) ♭ + adjNatInC' {c'} {c} {d} f h = + h ⋆⟨ C ⟩ (f ♭) + ≡⟨ sym (adjIso .rightInv (h ⋆⟨ C ⟩ (f ♭))) ⟩ + ((h ⋆⟨ C ⟩ (f ♭)) ♯) ♭ + ≡⟨ cong _♭ (adjNatInC (f ♭) h) ⟩ + ((F ⟪ h ⟫ ⋆⟨ D ⟩ (f ♭) ♯) ♭) + ≡⟨ cong _♭ (cong (λ f' → seq' D (F ⟪ h ⟫) f') (adjIso .leftInv f)) ⟩ + (F ⟪ h ⟫ ⋆⟨ D ⟩ f) ♭ ∎ + + isLeftAdjoint : {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) + isLeftAdjoint {C = C}{D} F = Σ[ G ∈ Functor D C ] F ⊣ G + + isRightAdjoint : {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (G : Functor D C) → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) + isRightAdjoint {C = C}{D} G = Σ[ F ∈ Functor C D ] F ⊣ G + +{- ============================================== Proofs of equivalence ============================================== @@ -123,160 +245,160 @@ The second unnamed module does the reverse. -} -module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) (G : Functor D C) where - open UnitCounit - open NaturalBijection renaming (_⊣_ to _⊣²_) - module _ (adj : F ⊣² G) where - open _⊣²_ adj - open _⊣_ - - -- Naturality condition implies that a commutative square in C - -- appears iff the transpose in D is commutative as well - -- Used in adj'→adj - adjNat' : ∀ {c c' d d'} {f : D [ F ⟅ c ⟆ , d ]} {k : D [ d , d' ]} - → {h : C [ c , c' ]} {g : C [ c' , G ⟅ d' ⟆ ]} - -- commutativity of squares is iff - → ((f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) → (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g)) - × ((f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g) → (f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯)) - adjNat' {c} {c'} {d} {d'} {f} {k} {h} {g} = D→C , C→D - where - D→C : (f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) → (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g) - D→C eq = f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ - ≡⟨ sym (adjNatInD _ _) ⟩ - ((f ⋆⟨ D ⟩ k) ♭) - ≡⟨ cong _♭ eq ⟩ - (F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) ♭ - ≡⟨ sym (cong _♭ (adjNatInC _ _)) ⟩ - (h ⋆⟨ C ⟩ g) ♯ ♭ - ≡⟨ adjIso .rightInv _ ⟩ - h ⋆⟨ C ⟩ g - ∎ - C→D : (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g) → (f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) - C→D eq = f ⋆⟨ D ⟩ k - ≡⟨ sym (adjIso .leftInv _) ⟩ - (f ⋆⟨ D ⟩ k) ♭ ♯ - ≡⟨ cong _♯ (adjNatInD _ _) ⟩ - (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ - ≡⟨ cong _♯ eq ⟩ - (h ⋆⟨ C ⟩ g) ♯ - ≡⟨ adjNatInC _ _ ⟩ - F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯ - ∎ - - open NatTrans - - -- note : had to make this record syntax because termination checker was complaining - -- due to referencing η and ε from the definitions of Δs - adj'→adj : F ⊣ G - adj'→adj = record - { η = η' - ; ε = ε' - ; triangleIdentities = record - {Δ₁ = Δ₁' - ; Δ₂ = Δ₂' }} - - where - -- ETA - - -- trivial commutative diagram between identities in D - commInD : ∀ {x y} (f : C [ x , y ]) → D .id ⋆⟨ D ⟩ F ⟪ f ⟫ ≡ F ⟪ f ⟫ ⋆⟨ D ⟩ D .id - commInD f = (D .⋆IdL _) ∙ sym (D .⋆IdR _) - - sharpen1 : ∀ {x y} (f : C [ x , y ]) → F ⟪ f ⟫ ⋆⟨ D ⟩ D .id ≡ F ⟪ f ⟫ ⋆⟨ D ⟩ D .id ♭ ♯ - sharpen1 f = cong (λ v → F ⟪ f ⟫ ⋆⟨ D ⟩ v) (sym (adjIso .leftInv _)) - - η' : 𝟙⟨ C ⟩ ⇒ G ∘F F - η' .N-ob x = D .id ♭ - η' .N-hom f = sym (fst (adjNat') (commInD f ∙ sharpen1 f)) - - -- EPSILON - - -- trivial commutative diagram between identities in C - commInC : ∀ {x y} (g : D [ x , y ]) → C .id ⋆⟨ C ⟩ G ⟪ g ⟫ ≡ G ⟪ g ⟫ ⋆⟨ C ⟩ C .id - commInC g = (C .⋆IdL _) ∙ sym (C .⋆IdR _) - - sharpen2 : ∀ {x y} (g : D [ x , y ]) → C .id ♯ ♭ ⋆⟨ C ⟩ G ⟪ g ⟫ ≡ C .id ⋆⟨ C ⟩ G ⟪ g ⟫ - sharpen2 g = cong (λ v → v ⋆⟨ C ⟩ G ⟪ g ⟫) (adjIso .rightInv _) - - ε' : F ∘F G ⇒ 𝟙⟨ D ⟩ - ε' .N-ob x = C .id ♯ - ε' .N-hom g = sym (snd adjNat' (sharpen2 g ∙ commInC g)) - - -- DELTA 1 - - Δ₁' : ∀ c → F ⟪ η' ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε' ⟦ F ⟅ c ⟆ ⟧ ≡ D .id - Δ₁' c = - F ⟪ η' ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε' ⟦ F ⟅ c ⟆ ⟧ - ≡⟨ sym (snd adjNat' (cong (λ v → (η' ⟦ c ⟧) ⋆⟨ C ⟩ v) (G .F-id))) ⟩ - D .id ⋆⟨ D ⟩ D .id - ≡⟨ D .⋆IdL _ ⟩ - D .id - ∎ - - -- DELTA 2 - - Δ₂' : ∀ d → η' ⟦ G ⟅ d ⟆ ⟧ ⋆⟨ C ⟩ G ⟪ ε' ⟦ d ⟧ ⟫ ≡ C .id - Δ₂' d = - (η' ⟦ G ⟅ d ⟆ ⟧) ⋆⟨ C ⟩ (G ⟪ ε' ⟦ d ⟧ ⟫) - ≡⟨ fst adjNat' (cong (λ v → v ⋆⟨ D ⟩ (ε' ⟦ d ⟧)) (sym (F .F-id))) ⟩ - C .id ⋆⟨ C ⟩ C .id - ≡⟨ C .⋆IdL _ ⟩ - C .id - ∎ - - - module _ (adj : F ⊣ G) where - open _⊣_ adj - open _⊣²_ - open NatTrans - - adj→adj' : F ⊣² G - -- ∀ {c d} → Iso (D [ F ⟅ c ⟆ , d ]) (C [ c , G ⟅ d ⟆ ]) - -- takes f to Gf precomposed with the unit - adj→adj' .adjIso {c = c} .fun f = η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ f ⟫ - -- takes g to Fg postcomposed with the counit - adj→adj' .adjIso {d = d} .inv g = F ⟪ g ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ - -- invertibility follows from the triangle identities - adj→adj' .adjIso {c = c} {d} .rightInv g - = η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ ⟫ - ≡⟨ cong (λ v → η ⟦ c ⟧ ⋆⟨ C ⟩ v) (G .F-seq _ _) ⟩ - η ⟦ c ⟧ ⋆⟨ C ⟩ (G ⟪ F ⟪ g ⟫ ⟫ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫) - ≡⟨ sym (C .⋆Assoc _ _ _) ⟩ - η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⟫ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫ - -- apply naturality - ≡⟨ rCatWhisker {C = C} _ _ _ natu ⟩ - (g ⋆⟨ C ⟩ η ⟦ G ⟅ d ⟆ ⟧) ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫ - ≡⟨ C .⋆Assoc _ _ _ ⟩ - g ⋆⟨ C ⟩ (η ⟦ G ⟅ d ⟆ ⟧ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫) - ≡⟨ lCatWhisker {C = C} _ _ _ (Δ₂ d) ⟩ - g ⋆⟨ C ⟩ C .id - ≡⟨ C .⋆IdR _ ⟩ - g - ∎ - where - natu : η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⟫ ≡ g ⋆⟨ C ⟩ η ⟦ G ⟅ d ⟆ ⟧ - natu = sym (η .N-hom _) - adj→adj' .adjIso {c = c} {d} .leftInv f - = F ⟪ η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ - ≡⟨ cong (λ v → v ⋆⟨ D ⟩ ε ⟦ d ⟧) (F .F-seq _ _) ⟩ - F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ F ⟪ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ - ≡⟨ D .⋆Assoc _ _ _ ⟩ - F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ (F ⟪ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧) - -- apply naturality - ≡⟨ lCatWhisker {C = D} _ _ _ natu ⟩ - F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ (ε ⟦ F ⟅ c ⟆ ⟧ ⋆⟨ D ⟩ f) - ≡⟨ sym (D .⋆Assoc _ _ _) ⟩ - F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε ⟦ F ⟅ c ⟆ ⟧ ⋆⟨ D ⟩ f - -- apply triangle identity - ≡⟨ rCatWhisker {C = D} _ _ _ (Δ₁ c) ⟩ - D .id ⋆⟨ D ⟩ f - ≡⟨ D .⋆IdL _ ⟩ - f - ∎ - where - natu : F ⟪ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ ≡ ε ⟦ F ⟅ c ⟆ ⟧ ⋆⟨ D ⟩ f - natu = ε .N-hom _ - -- follows directly from functoriality - adj→adj' .adjNatInD {c = c} f k = cong (λ v → η ⟦ c ⟧ ⋆⟨ C ⟩ v) (G .F-seq _ _) ∙ (sym (C .⋆Assoc _ _ _)) - adj→adj' .adjNatInC {d = d} g h = cong (λ v → v ⋆⟨ D ⟩ ε ⟦ d ⟧) (F .F-seq _ _) ∙ D .⋆Assoc _ _ _ +module _ (F : Functor C D) (G : Functor D C) where + open UnitCounit + open NaturalBijection renaming (_⊣_ to _⊣²_) + module _ (adj : F ⊣² G) where + open _⊣²_ adj + open _⊣_ + + -- Naturality condition implies that a commutative square in C + -- appears iff the transpose in D is commutative as well + -- Used in adj'→adj + adjNat' : ∀ {c c' d d'} {f : D [ F ⟅ c ⟆ , d ]} {k : D [ d , d' ]} + → {h : C [ c , c' ]} {g : C [ c' , G ⟅ d' ⟆ ]} + -- commutativity of squares is iff + → ((f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) → (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g)) + × ((f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g) → (f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯)) + adjNat' {c} {c'} {d} {d'} {f} {k} {h} {g} = D→C , C→D + where + D→C : (f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) → (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g) + D→C eq = f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ + ≡⟨ sym (adjNatInD _ _) ⟩ + ((f ⋆⟨ D ⟩ k) ♭) + ≡⟨ cong _♭ eq ⟩ + (F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) ♭ + ≡⟨ sym (cong _♭ (adjNatInC _ _)) ⟩ + (h ⋆⟨ C ⟩ g) ♯ ♭ + ≡⟨ adjIso .rightInv _ ⟩ + h ⋆⟨ C ⟩ g + ∎ + C→D : (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫ ≡ h ⋆⟨ C ⟩ g) → (f ⋆⟨ D ⟩ k ≡ F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯) + C→D eq = f ⋆⟨ D ⟩ k + ≡⟨ sym (adjIso .leftInv _) ⟩ + (f ⋆⟨ D ⟩ k) ♭ ♯ + ≡⟨ cong _♯ (adjNatInD _ _) ⟩ + (f ♭ ⋆⟨ C ⟩ G ⟪ k ⟫) ♯ + ≡⟨ cong _♯ eq ⟩ + (h ⋆⟨ C ⟩ g) ♯ + ≡⟨ adjNatInC _ _ ⟩ + F ⟪ h ⟫ ⋆⟨ D ⟩ g ♯ + ∎ + + open NatTrans + + -- note : had to make this record syntax because termination checker was complaining + -- due to referencing η and ε from the definitions of Δs + adj'→adj : F ⊣ G + adj'→adj = record + { η = η' + ; ε = ε' + ; triangleIdentities = record + {Δ₁ = Δ₁' + ; Δ₂ = Δ₂' }} + + where + -- ETA + + -- trivial commutative diagram between identities in D + commInD : ∀ {x y} (f : C [ x , y ]) → D .id ⋆⟨ D ⟩ F ⟪ f ⟫ ≡ F ⟪ f ⟫ ⋆⟨ D ⟩ D .id + commInD f = (D .⋆IdL _) ∙ sym (D .⋆IdR _) + + sharpen1 : ∀ {x y} (f : C [ x , y ]) → F ⟪ f ⟫ ⋆⟨ D ⟩ D .id ≡ F ⟪ f ⟫ ⋆⟨ D ⟩ D .id ♭ ♯ + sharpen1 f = cong (λ v → F ⟪ f ⟫ ⋆⟨ D ⟩ v) (sym (adjIso .leftInv _)) + + η' : 𝟙⟨ C ⟩ ⇒ G ∘F F + η' .N-ob x = D .id ♭ + η' .N-hom f = sym (fst (adjNat') (commInD f ∙ sharpen1 f)) + + -- EPSILON + + -- trivial commutative diagram between identities in C + commInC : ∀ {x y} (g : D [ x , y ]) → C .id ⋆⟨ C ⟩ G ⟪ g ⟫ ≡ G ⟪ g ⟫ ⋆⟨ C ⟩ C .id + commInC g = (C .⋆IdL _) ∙ sym (C .⋆IdR _) + + sharpen2 : ∀ {x y} (g : D [ x , y ]) → C .id ♯ ♭ ⋆⟨ C ⟩ G ⟪ g ⟫ ≡ C .id ⋆⟨ C ⟩ G ⟪ g ⟫ + sharpen2 g = cong (λ v → v ⋆⟨ C ⟩ G ⟪ g ⟫) (adjIso .rightInv _) + + ε' : F ∘F G ⇒ 𝟙⟨ D ⟩ + ε' .N-ob x = C .id ♯ + ε' .N-hom g = sym (snd adjNat' (sharpen2 g ∙ commInC g)) + + -- DELTA 1 + + Δ₁' : ∀ c → F ⟪ η' ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε' ⟦ F ⟅ c ⟆ ⟧ ≡ D .id + Δ₁' c = + F ⟪ η' ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε' ⟦ F ⟅ c ⟆ ⟧ + ≡⟨ sym (snd adjNat' (cong (λ v → (η' ⟦ c ⟧) ⋆⟨ C ⟩ v) (G .F-id))) ⟩ + D .id ⋆⟨ D ⟩ D .id + ≡⟨ D .⋆IdL _ ⟩ + D .id + ∎ + + -- DELTA 2 + + Δ₂' : ∀ d → η' ⟦ G ⟅ d ⟆ ⟧ ⋆⟨ C ⟩ G ⟪ ε' ⟦ d ⟧ ⟫ ≡ C .id + Δ₂' d = + (η' ⟦ G ⟅ d ⟆ ⟧) ⋆⟨ C ⟩ (G ⟪ ε' ⟦ d ⟧ ⟫) + ≡⟨ fst adjNat' (cong (λ v → v ⋆⟨ D ⟩ (ε' ⟦ d ⟧)) (sym (F .F-id))) ⟩ + C .id ⋆⟨ C ⟩ C .id + ≡⟨ C .⋆IdL _ ⟩ + C .id + ∎ + + + module _ (adj : F ⊣ G) where + open _⊣_ adj + open _⊣²_ + open NatTrans + + adj→adj' : F ⊣² G + -- ∀ {c d} → Iso (D [ F ⟅ c ⟆ , d ]) (C [ c , G ⟅ d ⟆ ]) + -- takes f to Gf precomposed with the unit + adj→adj' .adjIso {c = c} .fun f = η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ f ⟫ + -- takes g to Fg postcomposed with the counit + adj→adj' .adjIso {d = d} .inv g = F ⟪ g ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ + -- invertibility follows from the triangle identities + adj→adj' .adjIso {c = c} {d} .rightInv g + = η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ ⟫ + ≡⟨ cong (λ v → η ⟦ c ⟧ ⋆⟨ C ⟩ v) (G .F-seq _ _) ⟩ + η ⟦ c ⟧ ⋆⟨ C ⟩ (G ⟪ F ⟪ g ⟫ ⟫ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫) + ≡⟨ sym (C .⋆Assoc _ _ _) ⟩ + η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⟫ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫ + -- apply naturality + ≡⟨ rCatWhisker {C = C} _ _ _ natu ⟩ + (g ⋆⟨ C ⟩ η ⟦ G ⟅ d ⟆ ⟧) ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫ + ≡⟨ C .⋆Assoc _ _ _ ⟩ + g ⋆⟨ C ⟩ (η ⟦ G ⟅ d ⟆ ⟧ ⋆⟨ C ⟩ G ⟪ ε ⟦ d ⟧ ⟫) + ≡⟨ lCatWhisker {C = C} _ _ _ (Δ₂ d) ⟩ + g ⋆⟨ C ⟩ C .id + ≡⟨ C .⋆IdR _ ⟩ + g + ∎ + where + natu : η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⟫ ≡ g ⋆⟨ C ⟩ η ⟦ G ⟅ d ⟆ ⟧ + natu = sym (η .N-hom _) + adj→adj' .adjIso {c = c} {d} .leftInv f + = F ⟪ η ⟦ c ⟧ ⋆⟨ C ⟩ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ + ≡⟨ cong (λ v → v ⋆⟨ D ⟩ ε ⟦ d ⟧) (F .F-seq _ _) ⟩ + F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ F ⟪ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ + ≡⟨ D .⋆Assoc _ _ _ ⟩ + F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ (F ⟪ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧) + -- apply naturality + ≡⟨ lCatWhisker {C = D} _ _ _ natu ⟩ + F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ (ε ⟦ F ⟅ c ⟆ ⟧ ⋆⟨ D ⟩ f) + ≡⟨ sym (D .⋆Assoc _ _ _) ⟩ + F ⟪ η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ ε ⟦ F ⟅ c ⟆ ⟧ ⋆⟨ D ⟩ f + -- apply triangle identity + ≡⟨ rCatWhisker {C = D} _ _ _ (Δ₁ c) ⟩ + D .id ⋆⟨ D ⟩ f + ≡⟨ D .⋆IdL _ ⟩ + f + ∎ + where + natu : F ⟪ G ⟪ f ⟫ ⟫ ⋆⟨ D ⟩ ε ⟦ d ⟧ ≡ ε ⟦ F ⟅ c ⟆ ⟧ ⋆⟨ D ⟩ f + natu = ε .N-hom _ + -- follows directly from functoriality + adj→adj' .adjNatInD {c = c} f k = cong (λ v → η ⟦ c ⟧ ⋆⟨ C ⟩ v) (G .F-seq _ _) ∙ (sym (C .⋆Assoc _ _ _)) + adj→adj' .adjNatInC {d = d} g h = cong (λ v → v ⋆⟨ D ⟩ ε ⟦ d ⟧) (F .F-seq _ _) ∙ D .⋆Assoc _ _ _ \ No newline at end of file diff --git a/Cubical.Categories.Category.Base.html b/Cubical.Categories.Category.Base.html index 65f1e64575..1163e412d1 100644 --- a/Cubical.Categories.Category.Base.html +++ b/Cubical.Categories.Category.Base.html @@ -83,7 +83,7 @@ CatIso C x y = Σ[ f ∈ C [ x , y ] ] isIso C f CatIso≡ : {C : Category ℓ ℓ'}{x y : C .ob}(f g : CatIso C x y) → f .fst ≡ g .fst → f ≡ g -CatIso≡ f g = Σ≡Prop isPropIsIso +CatIso≡ f g = Σ≡Prop isPropIsIso -- `constructor` of CatIso catiso : {C : Category ℓ ℓ'}{x y : C .ob} @@ -137,22 +137,21 @@ ⋆Assoc (C ^op) f g h = sym (C .⋆Assoc _ _ _) isSetHom (C ^op) = C .isSetHom - -ΣPropCat : (C : Category ℓ ℓ') (P : ℙ (ob C)) → Category ℓ ℓ' -ob (ΣPropCat C P) = Σ[ x ∈ ob C ] x ∈ P -Hom[_,_] (ΣPropCat C P) x y = C [ fst x , fst y ] -id (ΣPropCat C P) = id C -_⋆_ (ΣPropCat C P) = _⋆_ C -⋆IdL (ΣPropCat C P) = ⋆IdL C -⋆IdR (ΣPropCat C P) = ⋆IdR C -⋆Assoc (ΣPropCat C P) = ⋆Assoc C -isSetHom (ΣPropCat C P) = isSetHom C - -isIsoΣPropCat : {C : Category ℓ ℓ'} {P : ℙ (ob C)} - {x y : ob C} (p : x ∈ P) (q : y ∈ P) - (f : C [ x , y ]) - → isIso C f → isIso (ΣPropCat C P) {x , p} {y , q} f -inv (isIsoΣPropCat p q f isIsoF) = isIsoF .inv -sec (isIsoΣPropCat p q f isIsoF) = isIsoF .sec -ret (isIsoΣPropCat p q f isIsoF) = isIsoF .ret +ΣPropCat : (C : Category ℓ ℓ') (P : ℙ (ob C)) → Category ℓ ℓ' +ob (ΣPropCat C P) = Σ[ x ∈ ob C ] x ∈ P +Hom[_,_] (ΣPropCat C P) x y = C [ fst x , fst y ] +id (ΣPropCat C P) = id C +_⋆_ (ΣPropCat C P) = _⋆_ C +⋆IdL (ΣPropCat C P) = ⋆IdL C +⋆IdR (ΣPropCat C P) = ⋆IdR C +⋆Assoc (ΣPropCat C P) = ⋆Assoc C +isSetHom (ΣPropCat C P) = isSetHom C + +isIsoΣPropCat : {C : Category ℓ ℓ'} {P : ℙ (ob C)} + {x y : ob C} (p : x ∈ P) (q : y ∈ P) + (f : C [ x , y ]) + → isIso C f → isIso (ΣPropCat C P) {x , p} {y , q} f +inv (isIsoΣPropCat p q f isIsoF) = isIsoF .inv +sec (isIsoΣPropCat p q f isIsoF) = isIsoF .sec +ret (isIsoΣPropCat p q f isIsoF) = isIsoF .ret \ No newline at end of file diff --git a/Cubical.Categories.Category.Properties.html b/Cubical.Categories.Category.Properties.html index ae39d91b42..71ac7f0c2d 100644 --- a/Cubical.Categories.Category.Properties.html +++ b/Cubical.Categories.Category.Properties.html @@ -92,4 +92,28 @@ → (r : PathP (λ i → C [ x , p i ]) f' f) → f ⋆⟨ C ⟩ g ≡ seqP' {p = p} f' g rCatWhiskerP f' f g r = cong (λ v → v ⋆⟨ C ⟩ g) (sym (fromPathP r)) + + + AssocCong₂⋆L : {x y' y z w : C .ob} → + {f' : C [ x , y' ]} {f : C [ x , y ]} + {g' : C [ y' , z ]} {g : C [ y , z ]} + → f ⋆⟨ C ⟩ g ≡ f' ⋆⟨ C ⟩ g' → (h : C [ z , w ]) + → f ⋆⟨ C ⟩ (g ⋆⟨ C ⟩ h) ≡ + f' ⋆⟨ C ⟩ (g' ⋆⟨ C ⟩ h) + AssocCong₂⋆L p h = + sym (⋆Assoc C _ _ h) + ∙∙ (λ i → p i ⋆⟨ C ⟩ h) ∙∙ + ⋆Assoc C _ _ h + + AssocCong₂⋆R : {x y z z' w : C .ob} → + (f : C [ x , y ]) + {g' : C [ y , z' ]} {g : C [ y , z ]} + {h' : C [ z' , w ]} {h : C [ z , w ]} + → g ⋆⟨ C ⟩ h ≡ g' ⋆⟨ C ⟩ h' + → (f ⋆⟨ C ⟩ g) ⋆⟨ C ⟩ h ≡ + (f ⋆⟨ C ⟩ g') ⋆⟨ C ⟩ h' + AssocCong₂⋆R f p = + ⋆Assoc C f _ _ + ∙∙ (λ i → f ⋆⟨ C ⟩ p i) ∙∙ + sym (⋆Assoc C _ _ _) \ No newline at end of file diff --git a/Cubical.Categories.Constructions.BinProduct.html b/Cubical.Categories.Constructions.BinProduct.html index 097299f3e9..110613873a 100644 --- a/Cubical.Categories.Constructions.BinProduct.html +++ b/Cubical.Categories.Constructions.BinProduct.html @@ -32,19 +32,19 @@ infixr 5 _×C_ -open Functor +open Functor -Fst : (C : Category ℓC ℓC') → (D : Category ℓD ℓD') → Functor (C ×C D) C -F-ob (Fst C D) = fst -F-hom (Fst C D) = fst -F-id (Fst C D) = refl -F-seq (Fst C D) _ _ = refl +Fst : (C : Category ℓC ℓC') → (D : Category ℓD ℓD') → Functor (C ×C D) C +F-ob (Fst C D) = fst +F-hom (Fst C D) = fst +F-id (Fst C D) = refl +F-seq (Fst C D) _ _ = refl -Snd : (C : Category ℓC ℓC') → (D : Category ℓD ℓD') → Functor (C ×C D) D -F-ob (Snd C D) = snd -F-hom (Snd C D) = snd -F-id (Snd C D) = refl -F-seq (Snd C D) _ _ = refl +Snd : (C : Category ℓC ℓC') → (D : Category ℓD ℓD') → Functor (C ×C D) D +F-ob (Snd C D) = snd +F-hom (Snd C D) = snd +F-id (Snd C D) = refl +F-seq (Snd C D) _ _ = refl module _ where private @@ -55,36 +55,36 @@ D : Category ℓD ℓD' E : Category ℓE ℓE' - open Functor + open Functor - _,F_ : Functor C D → Functor C E → Functor C (D ×C E) - (G ,F H) .F-ob a = (G ⟅ a ⟆ , H ⟅ a ⟆) - (G ,F H) .F-hom g = (G ⟪ g ⟫ , H ⟪ g ⟫) - (G ,F H) .F-id = ≡-× (G .F-id) (H .F-id) - (G ,F H) .F-seq _ _ = ≡-× (G .F-seq _ _) (H .F-seq _ _) + _,F_ : Functor C D → Functor C E → Functor C (D ×C E) + (G ,F H) .F-ob a = (G ⟅ a ⟆ , H ⟅ a ⟆) + (G ,F H) .F-hom g = (G ⟪ g ⟫ , H ⟪ g ⟫) + (G ,F H) .F-id = ≡-× (G .F-id) (H .F-id) + (G ,F H) .F-seq _ _ = ≡-× (G .F-seq _ _) (H .F-seq _ _) - _×F_ : Functor A C → Functor B D → Functor (A ×C B) (C ×C D) - _×F_ {A = A} {B = B} G H = G ∘F Fst A B ,F H ∘F Snd A B + _×F_ : Functor A C → Functor B D → Functor (A ×C B) (C ×C D) + _×F_ {A = A} {B = B} G H = G ∘F Fst A B ,F H ∘F Snd A B -- Some useful functors module _ (C : Category ℓC ℓC') (D : Category ℓD ℓD') where - open Functor + open Functor module _ (E : Category ℓE ℓE') where -- Associativity of product - ×C-assoc : Functor (C ×C (D ×C E)) ((C ×C D) ×C E) - ×C-assoc .F-ob (c , (d , e)) = ((c , d), e) - ×C-assoc .F-hom (f , (g , h)) = ((f , g), h) - ×C-assoc .F-id = refl - ×C-assoc .F-seq _ _ = refl + ×C-assoc : Functor (C ×C (D ×C E)) ((C ×C D) ×C E) + ×C-assoc .F-ob (c , (d , e)) = ((c , d), e) + ×C-assoc .F-hom (f , (g , h)) = ((f , g), h) + ×C-assoc .F-id = refl + ×C-assoc .F-seq _ _ = refl -- Left/right injections into product - linj : (d : ob D) → Functor C (C ×C D) - linj d = Id ,F Constant C D d + linj : (d : ob D) → Functor C (C ×C D) + linj d = Id ,F Constant C D d - rinj : (c : ob C) → Functor D (C ×C D) - rinj c = Constant D C c ,F Id + rinj : (c : ob C) → Functor D (C ×C D) + rinj c = Constant D C c ,F Id {- TODO: diff --git a/Cubical.Categories.Constructions.Elements.html b/Cubical.Categories.Constructions.Elements.html index a7288cb559..7410b27181 100644 --- a/Cubical.Categories.Constructions.Elements.html +++ b/Cubical.Categories.Constructions.Elements.html @@ -25,88 +25,88 @@ -- some issues -- * always need to specify objects during composition because can't infer isSet open Category -open Functor +open Functor module Covariant {ℓ ℓ'} {C : Category ℓ ℓ'} where - getIsSet : ∀ {ℓS} (F : Functor C (SET ℓS)) → (c : C .ob) → isSet (fst (F ⟅ c ⟆)) - getIsSet F c = snd (F ⟅ c ⟆) + getIsSet : ∀ {ℓS} (F : Functor C (SET ℓS)) → (c : C .ob) → isSet (fst (F ⟅ c ⟆)) + getIsSet F c = snd (F ⟅ c ⟆) - Element : ∀ {ℓS} (F : Functor C (SET ℓS)) → Type (ℓ-max ℓ ℓS) - Element F = Σ[ c ∈ C .ob ] fst (F ⟅ c ⟆) + Element : ∀ {ℓS} (F : Functor C (SET ℓS)) → Type (ℓ-max ℓ ℓS) + Element F = Σ[ c ∈ C .ob ] fst (F ⟅ c ⟆) infix 50 ∫_ - ∫_ : ∀ {ℓS} → Functor C (SET ℓS) → Category (ℓ-max ℓ ℓS) (ℓ-max ℓ' ℓS) + ∫_ : ∀ {ℓS} → Functor C (SET ℓS) → Category (ℓ-max ℓ ℓS) (ℓ-max ℓ' ℓS) -- objects are (c , x) pairs where c ∈ C and x ∈ F c (∫ F) .ob = Element F -- morphisms are f : c → c' which take x to x' - (∫ F) .Hom[_,_] (c , x) (c' , x') = fiber (λ (f : C [ c , c' ]) → (F ⟪ f ⟫) x) x' - (∫ F) .id {x = (c , x)} = C .id , funExt⁻ (F .F-id) x + (∫ F) .Hom[_,_] (c , x) (c' , x') = fiber (λ (f : C [ c , c' ]) → (F ⟪ f ⟫) x) x' + (∫ F) .id {x = (c , x)} = C .id , funExt⁻ (F .F-id) x (∫ F) ._⋆_ {c , x} {c₁ , x₁} {c₂ , x₂} (f , p) (g , q) - = (f ⋆⟨ C ⟩ g) , ((F ⟪ f ⋆⟨ C ⟩ g ⟫) x - ≡⟨ funExt⁻ (F .F-seq _ _) _ ⟩ - (F ⟪ g ⟫) ((F ⟪ f ⟫) x) - ≡⟨ cong (F ⟪ g ⟫) p ⟩ - (F ⟪ g ⟫) x₁ + = (f ⋆⟨ C ⟩ g) , ((F ⟪ f ⋆⟨ C ⟩ g ⟫) x + ≡⟨ funExt⁻ (F .F-seq _ _) _ ⟩ + (F ⟪ g ⟫) ((F ⟪ f ⟫) x) + ≡⟨ cong (F ⟪ g ⟫) p ⟩ + (F ⟪ g ⟫) x₁ ≡⟨ q ⟩ x₂ ∎) (∫ F) .⋆IdL o@{c , x} o1@{c' , x'} f'@(f , p) i - = (cIdL i) , isOfHLevel→isOfHLevelDep 1 (λ a → isS' ((F ⟪ a ⟫) x) x') p' p cIdL i + = (cIdL i) , isOfHLevel→isOfHLevelDep 1 (λ a → isS' ((F ⟪ a ⟫) x) x') p' p cIdL i where isS = getIsSet F c isS' = getIsSet F c' cIdL = C .⋆IdL f -- proof from composition with id - p' : (F ⟪ C .id ⋆⟨ C ⟩ f ⟫) x ≡ x' + p' : (F ⟪ C .id ⋆⟨ C ⟩ f ⟫) x ≡ x' p' = snd ((∫ F) ._⋆_ ((∫ F) .id) f') (∫ F) .⋆IdR o@{c , x} o1@{c' , x'} f'@(f , p) i - = (cIdR i) , isOfHLevel→isOfHLevelDep 1 (λ a → isS' ((F ⟪ a ⟫) x) x') p' p cIdR i + = (cIdR i) , isOfHLevel→isOfHLevelDep 1 (λ a → isS' ((F ⟪ a ⟫) x) x') p' p cIdR i where cIdR = C .⋆IdR f isS' = getIsSet F c' - p' : (F ⟪ f ⋆⟨ C ⟩ C .id ⟫) x ≡ x' + p' : (F ⟪ f ⋆⟨ C ⟩ C .id ⟫) x ≡ x' p' = snd ((∫ F) ._⋆_ f' ((∫ F) .id)) (∫ F) .⋆Assoc o@{c , x} o1@{c₁ , x₁} o2@{c₂ , x₂} o3@{c₃ , x₃} f'@(f , p) g'@(g , q) h'@(h , r) i - = (cAssoc i) , isOfHLevel→isOfHLevelDep 1 (λ a → isS₃ ((F ⟪ a ⟫) x) x₃) p1 p2 cAssoc i + = (cAssoc i) , isOfHLevel→isOfHLevelDep 1 (λ a → isS₃ ((F ⟪ a ⟫) x) x₃) p1 p2 cAssoc i where cAssoc = C .⋆Assoc f g h isS₃ = getIsSet F c₃ - p1 : (F ⟪ (f ⋆⟨ C ⟩ g) ⋆⟨ C ⟩ h ⟫) x ≡ x₃ + p1 : (F ⟪ (f ⋆⟨ C ⟩ g) ⋆⟨ C ⟩ h ⟫) x ≡ x₃ p1 = snd ((∫ F) ._⋆_ ((∫ F) ._⋆_ {o} {o1} {o2} f' g') h') - p2 : (F ⟪ f ⋆⟨ C ⟩ (g ⋆⟨ C ⟩ h) ⟫) x ≡ x₃ + p2 : (F ⟪ f ⋆⟨ C ⟩ (g ⋆⟨ C ⟩ h) ⟫) x ≡ x₃ p2 = snd ((∫ F) ._⋆_ f' ((∫ F) ._⋆_ {o1} {o2} {o3} g' h')) - (∫ F) .isSetHom {x} {y} = isSetΣSndProp (C .isSetHom) λ _ → (F ⟅ fst y ⟆) .snd _ _ + (∫ F) .isSetHom {x} {y} = isSetΣSndProp (C .isSetHom) λ _ → (F ⟅ fst y ⟆) .snd _ _ - ForgetElements : ∀ {ℓS} → (F : Functor C (SET ℓS)) → Functor (∫ F) C - F-ob (ForgetElements F) = fst - F-hom (ForgetElements F) = fst - F-id (ForgetElements F) = refl - F-seq (ForgetElements F) _ _ = refl + ForgetElements : ∀ {ℓS} → (F : Functor C (SET ℓS)) → Functor (∫ F) C + F-ob (ForgetElements F) = fst + F-hom (ForgetElements F) = fst + F-id (ForgetElements F) = refl + F-seq (ForgetElements F) _ _ = refl module Contravariant {ℓ ℓ'} {C : Category ℓ ℓ'} where open Covariant {C = C ^op} -- same thing but for presheaves - ∫ᴾ_ : ∀ {ℓS} → Functor (C ^op) (SET ℓS) → Category (ℓ-max ℓ ℓS) (ℓ-max ℓ' ℓS) + ∫ᴾ_ : ∀ {ℓS} → Functor (C ^op) (SET ℓS) → Category (ℓ-max ℓ ℓS) (ℓ-max ℓ' ℓS) ∫ᴾ F = (∫ F) ^op - Elementᴾ : ∀ {ℓS} → Functor (C ^op) (SET ℓS) → Type (ℓ-max ℓ ℓS) + Elementᴾ : ∀ {ℓS} → Functor (C ^op) (SET ℓS) → Type (ℓ-max ℓ ℓS) Elementᴾ F = (∫ᴾ F) .ob -- helpful results - module _ {ℓS} {F : Functor (C ^op) (SET ℓS)} where + module _ {ℓS} {F : Functor (C ^op) (SET ℓS)} where -- morphisms are equal as long as the morphisms in C are equal ∫ᴾhomEq : ∀ {o1 o1' o2 o2'} (f : (∫ᴾ F) [ o1 , o2 ]) (g : (∫ᴾ F) [ o1' , o2' ]) → (p : o1 ≡ o1') (q : o2 ≡ o2') → (eqInC : PathP (λ i → C [ fst (p i) , fst (q i) ]) (fst f) (fst g)) → PathP (λ i → (∫ᴾ F) [ p i , q i ]) f g - ∫ᴾhomEq _ _ _ _ = ΣPathPProp (λ f → snd (F ⟅ _ ⟆) _ _) + ∫ᴾhomEq _ _ _ _ = ΣPathPProp (λ f → snd (F ⟅ _ ⟆) _ _) ∫ᴾhomEqSimpl : ∀ {o1 o2} (f g : (∫ᴾ F) [ o1 , o2 ]) → fst f ≡ fst g → f ≡ g diff --git a/Cubical.Categories.Constructions.EssentialImage.html b/Cubical.Categories.Constructions.EssentialImage.html index 26bd2f0796..6e7865adda 100644 --- a/Cubical.Categories.Constructions.EssentialImage.html +++ b/Cubical.Categories.Constructions.EssentialImage.html @@ -22,14 +22,14 @@ module _ {C : Category ℓC ℓC'}{D : Category ℓD ℓD'} - (F : Functor C D) where + (F : Functor C D) where open Category - open Functor + open Functor isInEssentialImage : D .ob → Type (ℓ-max ℓC ℓD') - isInEssentialImage d = ∃[ c ∈ C .ob ] CatIso D (F .F-ob c) d + isInEssentialImage d = ∃[ c ∈ C .ob ] CatIso D (F .F-ob c) d isPropIsInEssentialImage : (d : D .ob) → isProp (isInEssentialImage d) isPropIsInEssentialImage _ = squash₁ @@ -40,27 +40,27 @@ EssentialImage = FullSubcategory D isInEssentialImage - ToEssentialImage : Functor C EssentialImage - ToEssentialImage .F-ob c = F .F-ob c , ∣ c , idCatIso ∣₁ - ToEssentialImage .F-hom = F .F-hom - ToEssentialImage .F-id = F .F-id - ToEssentialImage .F-seq = F .F-seq + ToEssentialImage : Functor C EssentialImage + ToEssentialImage .F-ob c = F .F-ob c , ∣ c , idCatIso ∣₁ + ToEssentialImage .F-hom = F .F-hom + ToEssentialImage .F-id = F .F-id + ToEssentialImage .F-seq = F .F-seq - FromEssentialImage : Functor EssentialImage D + FromEssentialImage : Functor EssentialImage D FromEssentialImage = FullInclusion D isInEssentialImage - CompEssentialImage : funcComp FromEssentialImage ToEssentialImage ≡ F - CompEssentialImage = Functor≡ (λ _ → refl) (λ _ → refl) + CompEssentialImage : funcComp FromEssentialImage ToEssentialImage ≡ F + CompEssentialImage = Functor≡ (λ _ → refl) (λ _ → refl) - isEssentiallySurjToEssentialImage : isEssentiallySurj ToEssentialImage + isEssentiallySurjToEssentialImage : isEssentiallySurj ToEssentialImage isEssentiallySurjToEssentialImage (d , p) = PropTrunc.map (λ (c , f) → c , Incl-Iso-inv _ _ _ _ f) p - isFullyFaithfulFromEssentialImage : isFullyFaithful FromEssentialImage + isFullyFaithfulFromEssentialImage : isFullyFaithful FromEssentialImage isFullyFaithfulFromEssentialImage = isFullyFaithfulIncl D isInEssentialImage - isFullyFaithfulToEssentialImage : isFullyFaithful F → isFullyFaithful ToEssentialImage + isFullyFaithfulToEssentialImage : isFullyFaithful F → isFullyFaithful ToEssentialImage isFullyFaithfulToEssentialImage fullfaith = fullfaith diff --git a/Cubical.Categories.Constructions.Free.Category.Base.html b/Cubical.Categories.Constructions.Free.Category.Base.html index b1a0be6a95..4d7fea8e06 100644 --- a/Cubical.Categories.Constructions.Free.Category.Base.html +++ b/Cubical.Categories.Constructions.Free.Category.Base.html @@ -26,7 +26,7 @@ ℓc ℓc' ℓd ℓd' ℓg ℓg' : Level open Category -open Functor +open Functor open NatIso hiding (sqRL; sqLL) open NatTrans @@ -55,7 +55,7 @@ η ._$g_ = λ z → z η ._<$g>_ = ↑_ - module _ {ℓc ℓc'} {𝓒 : Category ℓc ℓc'} (F F' : Functor FreeCat 𝓒) where + module _ {ℓc ℓc'} {𝓒 : Category ℓc ℓc'} (F F' : Functor FreeCat 𝓒) where -- Formulating uniqueness this way works out best definitionally. -- If you prove induction from the alternative below of @@ -63,20 +63,20 @@ -- then you have to use path comp which has bad definitional behavior module _ (agree-on-η : F ∘Interp η ≡ F' ∘Interp η) where private - aoo : ∀ c → F ⟅ c ⟆ ≡ F' ⟅ c ⟆ + aoo : ∀ c → F ⟅ c ⟆ ≡ F' ⟅ c ⟆ aoo = (λ c i → agree-on-η i $g c) aom-t : ∀ {c c'} (e : Exp c c') → Type _ aom-t {c}{c'} e = - PathP (λ i → 𝓒 [ aoo c i , aoo c' i ]) (F ⟪ e ⟫) (F' ⟪ e ⟫) + PathP (λ i → 𝓒 [ aoo c i , aoo c' i ]) (F ⟪ e ⟫) (F' ⟪ e ⟫) aom-id : ∀ {c} → aom-t (idₑ {c}) - aom-id = F .F-id ◁ (λ i → 𝓒 .id) ▷ sym (F' .F-id) + aom-id = F .F-id ◁ (λ i → 𝓒 .id) ▷ sym (F' .F-id) aom-seq : ∀ {c c' c''} (e : Exp c c')(e' : Exp c' c'') → aom-t e → aom-t e' → aom-t (e ⋆ₑ e') aom-seq e e' ihe ihe' = - F .F-seq e e' ◁ (λ i → ihe i ⋆⟨ 𝓒 ⟩ ihe' i) ▷ sym (F' .F-seq e e') + F .F-seq e e' ◁ (λ i → ihe i ⋆⟨ 𝓒 ⟩ ihe' i) ▷ sym (F' .F-seq e e') aom : ∀ {c c'} (e : Exp c c') → aom-t e aom (↑ x) = λ i → agree-on-η i <$g> x @@ -87,27 +87,27 @@ (λ i j → 𝓒 .isSetHom) (aom-seq idₑ e aom-id (aom e)) (aom e) - (λ i → F ⟪ ⋆ₑIdL e i ⟫) ((λ i → F' ⟪ ⋆ₑIdL e i ⟫)) i + (λ i → F ⟪ ⋆ₑIdL e i ⟫) ((λ i → F' ⟪ ⋆ₑIdL e i ⟫)) i aom (⋆ₑIdR e i) = isSet→SquareP (λ i j → 𝓒 .isSetHom) (aom-seq e idₑ (aom e) aom-id) (aom e) - (λ i → F ⟪ ⋆ₑIdR e i ⟫) ((λ i → F' ⟪ ⋆ₑIdR e i ⟫)) i + (λ i → F ⟪ ⋆ₑIdR e i ⟫) ((λ i → F' ⟪ ⋆ₑIdR e i ⟫)) i aom (⋆ₑAssoc e e' e'' i) = isSet→SquareP (λ _ _ → 𝓒 .isSetHom) (aom-seq _ _ (aom-seq _ _ (aom e) (aom e')) (aom e'')) (aom-seq _ _ (aom e) (aom-seq _ _ (aom e') (aom e''))) - ((λ i → F ⟪ ⋆ₑAssoc e e' e'' i ⟫)) - (λ i → F' ⟪ ⋆ₑAssoc e e' e'' i ⟫) + ((λ i → F ⟪ ⋆ₑAssoc e e' e'' i ⟫)) + (λ i → F' ⟪ ⋆ₑAssoc e e' e'' i ⟫) i aom (isSetExp e e' x y i j) = isSet→SquareP {A = λ i j → aom-t (isSetExp e e' x y i j)} (λ i j → isOfHLevelPathP 2 (𝓒 .isSetHom) - (F ⟪ isSetExp e e' x y i j ⟫) - (F' ⟪ isSetExp e e' x y i j ⟫)) + (F ⟪ isSetExp e e' x y i j ⟫) + (F' ⟪ isSetExp e e' x y i j ⟫)) (λ j → aom (x j)) (λ j → aom (y j)) (λ i → aom e) @@ -115,7 +115,7 @@ i j induction : F ≡ F' - induction = Functor≡ aoo aom + induction = Functor≡ aoo aom module Semantics {ℓc ℓc'} (𝓒 : Category ℓc ℓc') @@ -130,19 +130,19 @@ ⟦ isSetExp e e' p q i j ⟧ = 𝓒 .isSetHom ⟦ e ⟧ ⟦ e' ⟧ (cong ⟦_⟧ p) (cong ⟦_⟧ q) i j - sem : Functor FreeCat 𝓒 - sem .Functor.F-ob v = ı $g v - sem .Functor.F-hom e = ⟦ e ⟧ - sem .Functor.F-id = refl - sem .Functor.F-seq e e' = refl + sem : Functor FreeCat 𝓒 + sem .Functor.F-ob v = ı $g v + sem .Functor.F-hom e = ⟦ e ⟧ + sem .Functor.F-id = refl + sem .Functor.F-seq e e' = refl sem-extends-ı : (η ⋆Interp sem) ≡ ı sem-extends-ı = refl - sem-uniq : ∀ {F : Functor FreeCat 𝓒} → ((Functor→GraphHom F ∘GrHom η) ≡ ı) → F ≡ sem + sem-uniq : ∀ {F : Functor FreeCat 𝓒} → ((Functor→GraphHom F ∘GrHom η) ≡ ı) → F ≡ sem sem-uniq {F} aog = induction F sem aog - sem-contr : ∃![ F ∈ Functor FreeCat 𝓒 ] Functor→GraphHom F ∘GrHom η ≡ ı + sem-contr : ∃![ F ∈ Functor FreeCat 𝓒 ] Functor→GraphHom F ∘GrHom η ≡ ı sem-contr .fst = sem , sem-extends-ı sem-contr .snd (sem' , sem'-extends-ı) = ΣPathP paths where @@ -153,18 +153,18 @@ paths .fst = sym (sem-uniq sem'-extends-ı) paths .snd i j = sem'-extends-ı ((~ i) ∨ j) - η-expansion : {𝓒 : Category ℓc ℓc'} (F : Functor FreeCat 𝓒) + η-expansion : {𝓒 : Category ℓc ℓc'} (F : Functor FreeCat 𝓒) → F ≡ Semantics.sem 𝓒 (F ∘Interp η) η-expansion {𝓒 = 𝓒} F = induction F (Semantics.sem 𝓒 (F ∘Interp η)) refl -- co-unit of the 2-adjunction module _ {𝓒 : Category ℓc ℓc'} where open FreeCategory (Cat→Graph 𝓒) - ε : Functor FreeCat 𝓒 - ε = Semantics.sem 𝓒 (Functor→GraphHom {𝓓 = 𝓒} Id) + ε : Functor FreeCat 𝓒 + ε = Semantics.sem 𝓒 (Functor→GraphHom {𝓓 = 𝓒} Id) ε-reasoning : {𝓓 : Category ℓd ℓd'} - → (𝓕 : Functor 𝓒 𝓓) - → 𝓕 ∘F ε ≡ Semantics.sem 𝓓 (Functor→GraphHom 𝓕) + → (𝓕 : Functor 𝓒 𝓓) + → 𝓕 ∘F ε ≡ Semantics.sem 𝓓 (Functor→GraphHom 𝓕) ε-reasoning {𝓓 = 𝓓} 𝓕 = Semantics.sem-uniq 𝓓 (Functor→GraphHom 𝓕) refl \ No newline at end of file diff --git a/Cubical.Categories.Constructions.Free.Functor.html b/Cubical.Categories.Constructions.Free.Functor.html index 1b8e8a7c7f..b9d2628806 100644 --- a/Cubical.Categories.Constructions.Free.Functor.html +++ b/Cubical.Categories.Constructions.Free.Functor.html @@ -29,7 +29,7 @@ ℓ ℓ' ℓc ℓc' ℓd ℓd' ℓg ℓg' ℓh ℓh' : Level open Category -open Functor +open Functor open NatTrans open NatIso open isIso @@ -69,11 +69,11 @@ FH .⋆Assoc = ⋆ₑAssoc FH .isSetHom = isSetFExp - Fϕ : Functor FG FH - Fϕ .F-ob = ϕ - Fϕ .F-hom = F⟪_⟫ - Fϕ .F-id = F-idₑ - Fϕ .F-seq = F-seqₑ + Fϕ : Functor FG FH + Fϕ .F-ob = ϕ + Fϕ .F-hom = F⟪_⟫ + Fϕ .F-id = F-idₑ + Fϕ .F-seq = F-seqₑ -- The universal interpretation ηG = FreeCatG.η @@ -86,10 +86,10 @@ Fϕ-homo $g x = ϕ x Fϕ-homo <$g> x = F⟪ ↑ x ⟫ - ηϕ : Eq (Fϕ .F-ob ∘f ηG ._$g_) (ηH ._$g_ ∘f ϕ) + ηϕ : Eq (Fϕ .F-ob ∘f ηG ._$g_) (ηH ._$g_ ∘f ϕ) ηϕ = reflEq - module _ {𝓒 : Category ℓc ℓc'}{𝓓 : Category ℓd ℓd'} {𝓕 : Functor 𝓒 𝓓} where + module _ {𝓒 : Category ℓc ℓc'}{𝓓 : Category ℓd ℓd'} {𝓕 : Functor 𝓒 𝓓} where {- It is very important for the implementation of the functor @@ -101,7 +101,7 @@ -} module Semantics (ıG : Interp G 𝓒) (ıH : Interp H 𝓓) - (ıϕ : Eq (𝓕 .F-ob ∘f ıG ._$g_) (ıH ._$g_ ∘f ϕ)) + (ıϕ : Eq (𝓕 .F-ob ∘f ıG ._$g_) (ıH ._$g_ ∘f ϕ)) where semG = FreeCatG.Semantics.sem 𝓒 ıG @@ -112,34 +112,34 @@ semH-hom (F⟪_⟫ {A}{B} x) = transportEq (λ (f : G .Node → 𝓓 .ob) → 𝓓 [ f A , f B ]) ıϕ - (𝓕 ⟪ semG ⟪ x ⟫ ⟫) + (𝓕 ⟪ semG ⟪ x ⟫ ⟫) semH-hom (⋆ₑIdL f i) = 𝓓 .⋆IdL (semH-hom f) i semH-hom (⋆ₑIdR f i) = 𝓓 .⋆IdR (semH-hom f) i semH-hom (⋆ₑAssoc f f' f'' i) = 𝓓 .⋆Assoc (semH-hom f) (semH-hom f') (semH-hom f'') i semH-hom (F-idₑ {A} i) = unbound i where - unbound : transportEq (λ f → 𝓓 [ f A , f A ]) ıϕ (𝓕 ⟪ semG ⟪ idₑ ⟫ ⟫) + unbound : transportEq (λ f → 𝓓 [ f A , f A ]) ıϕ (𝓕 ⟪ semG ⟪ idₑ ⟫ ⟫) ≡ 𝓓 .id unbound = J (λ g p → transportEq (λ f → 𝓓 [ f A , f A ]) p - (𝓕 ⟪ semG ⟪ idₑ ⟫ ⟫) + (𝓕 ⟪ semG ⟪ idₑ ⟫ ⟫) ≡ 𝓓 .id) - ((𝓕 ∘F semG) .F-id) ıϕ + ((𝓕 ∘F semG) .F-id) ıϕ semH-hom (F-seqₑ {A}{B}{C} e e' i) = unbound i where unbound : - transportEq (λ f → 𝓓 [ f A , f C ]) ıϕ (𝓕 ⟪ semG ⟪ e ⋆ₑ e' ⟫ ⟫) - ≡ (transportEq (λ f → 𝓓 [ f A , f B ]) ıϕ (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) + transportEq (λ f → 𝓓 [ f A , f C ]) ıϕ (𝓕 ⟪ semG ⟪ e ⋆ₑ e' ⟫ ⟫) + ≡ (transportEq (λ f → 𝓓 [ f A , f B ]) ıϕ (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) ⋆⟨ 𝓓 ⟩ (transportEq (λ f → 𝓓 [ f B , f C ]) ıϕ - (𝓕 ⟪ semG ⟪ e' ⟫ ⟫)) + (𝓕 ⟪ semG ⟪ e' ⟫ ⟫)) unbound = J (λ g p → - transportEq (λ f → 𝓓 [ f A , f C ]) p (𝓕 ⟪ semG ⟪ e ⋆ₑ e' ⟫ ⟫) - ≡ (transportEq (λ f → 𝓓 [ f A , f B ]) p (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) + transportEq (λ f → 𝓓 [ f A , f C ]) p (𝓕 ⟪ semG ⟪ e ⋆ₑ e' ⟫ ⟫) + ≡ (transportEq (λ f → 𝓓 [ f A , f B ]) p (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) ⋆⟨ 𝓓 ⟩ (transportEq (λ f → 𝓓 [ f B , f C ]) p - (𝓕 ⟪ semG ⟪ e' ⟫ ⟫))) - ((𝓕 ∘F semG) .F-seq e e') + (𝓕 ⟪ semG ⟪ e' ⟫ ⟫))) + ((𝓕 ∘F semG) .F-seq e e') ıϕ semH-hom (isSetFExp f g p q i j) = 𝓓 .isSetHom (semH-hom f) @@ -149,18 +149,18 @@ i j - semH : Functor FH 𝓓 - semH .F-ob = ıH ._$g_ - semH .F-hom = semH-hom - semH .F-id = refl - semH .F-seq f g = refl + semH : Functor FH 𝓓 + semH .F-ob = ıH ._$g_ + semH .F-hom = semH-hom + semH .F-id = refl + semH .F-seq f g = refl - semϕ : Eq (𝓕 ∘F semG) (semH ∘F Fϕ) - semϕ = pathToEq (FreeCatG.induction (funcComp 𝓕 semG) - (funcComp semH Fϕ) + semϕ : Eq (𝓕 ∘F semG) (semH ∘F Fϕ) + semϕ = pathToEq (FreeCatG.induction (funcComp 𝓕 semG) + (funcComp semH Fϕ) (GrHom≡ aoo aoe)) where - 𝓕G = (𝓕 .F-ob ∘f ıG ._$g_) + 𝓕G = (𝓕 .F-ob ∘f ıG ._$g_) Hϕ = (ıH ._$g_ ∘f ϕ) aoo-gen : ∀ (v : Node G) f g @@ -168,125 +168,125 @@ → Path _ (f v) (g v) aoo-gen v f g = J ((λ f' _ → Path _ (f v) (f' v))) refl aoo : (v : Node G) - → Path _ (((𝓕 ∘F semG) ∘Interp ηG) $g v) - (((semH ∘F Fϕ) ∘Interp ηG) $g v) + → Path _ (((𝓕 ∘F semG) ∘Interp ηG) $g v) + (((semH ∘F Fϕ) ∘Interp ηG) $g v) aoo v = aoo-gen v 𝓕G Hϕ ıϕ aoe : {v w : Node G} (e : G .Edge v w) → PathP (λ i → 𝓓 .Hom[_,_] (aoo v i) (aoo w i)) - (𝓕 ⟪ semG ⟪ ↑ e ⟫ ⟫) - (semH ⟪ Fϕ ⟪ ↑ e ⟫ ⟫) + (𝓕 ⟪ semG ⟪ ↑ e ⟫ ⟫) + (semH ⟪ Fϕ ⟪ ↑ e ⟫ ⟫) aoe {v}{w} e = toPathP lem where lem : Path _ (transport (λ i → 𝓓 [ aoo-gen v 𝓕G Hϕ ıϕ i , aoo-gen w 𝓕G Hϕ ıϕ i ]) - (𝓕 ⟪ ıG <$g> e ⟫)) - (transportEq (λ f → 𝓓 [ f v , f w ]) ıϕ (𝓕 ⟪ ıG <$g> e ⟫)) + (𝓕 ⟪ ıG <$g> e ⟫)) + (transportEq (λ f → 𝓓 [ f v , f w ]) ıϕ (𝓕 ⟪ ıG <$g> e ⟫)) lem = J (λ f p → Path _ ((transport (λ i → 𝓓 [ aoo-gen v 𝓕G f p i , aoo-gen w 𝓕G f p i ]) - (𝓕 ⟪ ıG <$g> e ⟫))) + (𝓕 ⟪ ıG <$g> e ⟫))) ((transportEq (λ f → 𝓓 [ f v , f w ]) p - (𝓕 ⟪ ıG <$g> e ⟫)))) - (transportRefl (𝓕 ⟪ ıG <$g> e ⟫)) + (𝓕 ⟪ ıG <$g> e ⟫)))) + (transportRefl (𝓕 ⟪ ıG <$g> e ⟫)) ıϕ - module Uniqueness (arb𝓒 : Functor FG 𝓒) - (arb𝓓 : Functor FH 𝓓) - (arb𝓕 : Path (Functor FG 𝓓) (𝓕 ∘F arb𝓒) (arb𝓓 ∘F Fϕ)) + module Uniqueness (arb𝓒 : Functor FG 𝓒) + (arb𝓓 : Functor FH 𝓓) + (arb𝓕 : Path (Functor FG 𝓓) (𝓕 ∘F arb𝓒) (arb𝓓 ∘F Fϕ)) (arb𝓒-agree : arb𝓒 ∘Interp ηG ≡ ıG) (arb𝓓-agree : arb𝓓 ∘Interp ηH ≡ ıH) (arb𝓕-agree : Square {A = G .Node → 𝓓 .ob} - (λ i x → arb𝓕 i ⟅ x ⟆) + (λ i x → arb𝓕 i ⟅ x ⟆) (eqToPath ıϕ) - (λ i x → 𝓕 ⟅ arb𝓒-agree i $g x ⟆) + (λ i x → 𝓕 ⟅ arb𝓒-agree i $g x ⟆) (λ i x → arb𝓓-agree i $g (ϕ x))) where sem-uniq-G : arb𝓒 ≡ semG sem-uniq-G = FreeCatG.Semantics.sem-uniq _ _ arb𝓒-agree sem-uniq-H : arb𝓓 ≡ semH - sem-uniq-H = Functor≡ aoo aom where - aoo : (v : H .Node) → arb𝓓 ⟅ v ⟆ ≡ ıH $g v + sem-uniq-H = Functor≡ aoo aom where + aoo : (v : H .Node) → arb𝓓 ⟅ v ⟆ ≡ ıH $g v aoo = (λ v i → arb𝓓-agree i $g v) aom-type : ∀ {v w} → (f : FH [ v , w ]) → Type _ aom-type {v}{w} f = PathP (λ i → 𝓓 [ aoo v i , aoo w i ]) - (arb𝓓 ⟪ f ⟫) - (semH ⟪ f ⟫) + (arb𝓓 ⟪ f ⟫) + (semH ⟪ f ⟫) aom-id : ∀ {v} → aom-type {v} idₑ - aom-id = arb𝓓 .F-id ◁ λ i → 𝓓 .id + aom-id = arb𝓓 .F-id ◁ λ i → 𝓓 .id aom-seq : ∀ {v w x} → {f : FH [ v , w ]} {g : FH [ w , x ]} → aom-type f → aom-type g → aom-type (f ⋆ₑ g) - aom-seq hypf hypg = arb𝓓 .F-seq _ _ ◁ λ i → hypf i ⋆⟨ 𝓓 ⟩ hypg i + aom-seq hypf hypg = arb𝓓 .F-seq _ _ ◁ λ i → hypf i ⋆⟨ 𝓓 ⟩ hypg i ıϕp = eqToPath ıϕ aom-F : ∀ {v w} → (e : FG [ v , w ]) → PathP (λ i → 𝓓 [ (arb𝓓-agree i $g (ϕ v)) , (arb𝓓-agree i $g (ϕ w)) ]) - (arb𝓓 ⟪ Fϕ ⟪ e ⟫ ⟫) + (arb𝓓 ⟪ Fϕ ⟪ e ⟫ ⟫) (transportEq (λ (f : G .Node → 𝓓 .ob) → 𝓓 [ f v , f w ]) ıϕ - (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) + (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) aom-F {v}{w} e = pathified ▷ eqToPath ( substPath≡transport' (λ (f : G .Node → 𝓓 .ob) → 𝓓 [ f v , f w ]) - (𝓕 ⟪ semG ⟪ e ⟫ ⟫) + (𝓕 ⟪ semG ⟪ e ⟫ ⟫) ıϕ) where pathified : PathP (λ i → 𝓓 [ arb𝓓-agree i $g ϕ v , arb𝓓-agree i $g ϕ w ]) - (arb𝓓 ⟪ Fϕ ⟪ e ⟫ ⟫) + (arb𝓓 ⟪ Fϕ ⟪ e ⟫ ⟫) (transport (λ i → 𝓓 [ ıϕp i v , ıϕp i w ]) - (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) + (𝓕 ⟪ semG ⟪ e ⟫ ⟫)) pathified = toPathP⁻ (( fromPathP⁻ lem' - ∙ cong (transport⁻ (λ i → 𝓓 [ arb𝓕 (~ i) ⟅ v ⟆ - , arb𝓕 (~ i) ⟅ w ⟆ ])) + ∙ cong (transport⁻ (λ i → 𝓓 [ arb𝓕 (~ i) ⟅ v ⟆ + , arb𝓕 (~ i) ⟅ w ⟆ ])) (fromPathP⁻ lem𝓒) ∙ sym (transportComposite - ((λ i → 𝓓 [ 𝓕 ⟅ arb𝓒-agree (~ i) $g v ⟆ - , 𝓕 ⟅ arb𝓒-agree (~ i) $g w ⟆ ])) - (λ i → 𝓓 [ arb𝓕 i ⟅ v ⟆ , arb𝓕 i ⟅ w ⟆ ]) - ((𝓕 ⟪ semG ⟪ e ⟫ ⟫))) - ∙ ((λ i → transport (substOf-sems-agreeϕ i) (𝓕 ⟪ semG ⟪ e ⟫ ⟫))) + ((λ i → 𝓓 [ 𝓕 ⟅ arb𝓒-agree (~ i) $g v ⟆ + , 𝓕 ⟅ arb𝓒-agree (~ i) $g w ⟆ ])) + (λ i → 𝓓 [ arb𝓕 i ⟅ v ⟆ , arb𝓕 i ⟅ w ⟆ ]) + ((𝓕 ⟪ semG ⟪ e ⟫ ⟫))) + ∙ ((λ i → transport (substOf-sems-agreeϕ i) (𝓕 ⟪ semG ⟪ e ⟫ ⟫))) ∙ transportComposite (λ i → 𝓓 [ ıϕp i v , ıϕp i w ]) (λ i → 𝓓 [ arb𝓓-agree (~ i) $g ϕ v , arb𝓓-agree (~ i) $g ϕ w ]) - (𝓕 ⟪ semG ⟪ e ⟫ ⟫) + (𝓕 ⟪ semG ⟪ e ⟫ ⟫) )) where - lem' : PathP (λ i → 𝓓 [ arb𝓕 (~ i) ⟅ v ⟆ - , arb𝓕 (~ i) ⟅ w ⟆ ]) - (arb𝓓 ⟪ Fϕ ⟪ e ⟫ ⟫) - (𝓕 ⟪ arb𝓒 ⟪ e ⟫ ⟫) - lem' i = arb𝓕 (~ i) ⟪ e ⟫ + lem' : PathP (λ i → 𝓓 [ arb𝓕 (~ i) ⟅ v ⟆ + , arb𝓕 (~ i) ⟅ w ⟆ ]) + (arb𝓓 ⟪ Fϕ ⟪ e ⟫ ⟫) + (𝓕 ⟪ arb𝓒 ⟪ e ⟫ ⟫) + lem' i = arb𝓕 (~ i) ⟪ e ⟫ - lem𝓒 : PathP (λ i → 𝓓 [ 𝓕 ⟅ arb𝓒-agree i $g v ⟆ - , 𝓕 ⟅ arb𝓒-agree i $g w ⟆ ]) - (𝓕 ⟪ arb𝓒 ⟪ e ⟫ ⟫) - (𝓕 ⟪ semG ⟪ e ⟫ ⟫) - lem𝓒 i = 𝓕 ⟪ sem-uniq-G i ⟪ e ⟫ ⟫ + lem𝓒 : PathP (λ i → 𝓓 [ 𝓕 ⟅ arb𝓒-agree i $g v ⟆ + , 𝓕 ⟅ arb𝓒-agree i $g w ⟆ ]) + (𝓕 ⟪ arb𝓒 ⟪ e ⟫ ⟫) + (𝓕 ⟪ semG ⟪ e ⟫ ⟫) + lem𝓒 i = 𝓕 ⟪ sem-uniq-G i ⟪ e ⟫ ⟫ substOf-sems-agreeϕ : - ((λ i → 𝓓 [ 𝓕 ⟅ arb𝓒-agree (~ i) $g v ⟆ - , 𝓕 ⟅ arb𝓒-agree (~ i) $g w ⟆ ]) - ∙ (λ i → 𝓓 [ arb𝓕 i ⟅ v ⟆ , arb𝓕 i ⟅ w ⟆ ])) + ((λ i → 𝓓 [ 𝓕 ⟅ arb𝓒-agree (~ i) $g v ⟆ + , 𝓕 ⟅ arb𝓒-agree (~ i) $g w ⟆ ]) + ∙ (λ i → 𝓓 [ arb𝓕 i ⟅ v ⟆ , arb𝓕 i ⟅ w ⟆ ])) ≡ ((λ i → 𝓓 [ ıϕp i v , ıϕp i w ]) ∙ (λ i → 𝓓 [ arb𝓓-agree (~ i) $g ϕ v , arb𝓓-agree (~ i) $g ϕ w ])) substOf-sems-agreeϕ = - sym (cong-∙ A (λ i x → 𝓕 ⟅ arb𝓒-agree (~ i) $g x ⟆) - λ i x → arb𝓕 i ⟅ x ⟆) + sym (cong-∙ A (λ i x → 𝓕 ⟅ arb𝓒-agree (~ i) $g x ⟆) + λ i x → arb𝓕 i ⟅ x ⟆) ∙ cong (cong A) (Square→compPath λ i j x → arb𝓕-agree (~ i) j x) ∙ cong-∙ A (λ i x → ıϕp i x) @@ -303,46 +303,46 @@ (λ i j → 𝓓 .isSetHom) (aom-seq aom-id (aom f)) (aom f) - (λ i → arb𝓓 ⟪ ⋆ₑIdL f i ⟫) - (λ i → (semH ⟪ ⋆ₑIdL f i ⟫)) + (λ i → arb𝓓 ⟪ ⋆ₑIdL f i ⟫) + (λ i → (semH ⟪ ⋆ₑIdL f i ⟫)) i aom (⋆ₑIdR f i) = isSet→SquareP (λ i j → 𝓓 .isSetHom) (aom-seq (aom f) aom-id) (aom f ) - (λ i → arb𝓓 ⟪ ⋆ₑIdR f i ⟫) - (λ i → semH ⟪ ⋆ₑIdR f i ⟫) + (λ i → arb𝓓 ⟪ ⋆ₑIdR f i ⟫) + (λ i → semH ⟪ ⋆ₑIdR f i ⟫) i aom (⋆ₑAssoc f f₁ f₂ i) = isSet→SquareP (λ i j → 𝓓 .isSetHom) (aom-seq (aom-seq (aom f) (aom f₁)) (aom f₂)) (aom-seq (aom f) (aom-seq (aom f₁) (aom f₂))) - (λ i → arb𝓓 ⟪ ⋆ₑAssoc f f₁ f₂ i ⟫) - (λ i → semH ⟪ ⋆ₑAssoc f f₁ f₂ i ⟫) + (λ i → arb𝓓 ⟪ ⋆ₑAssoc f f₁ f₂ i ⟫) + (λ i → semH ⟪ ⋆ₑAssoc f f₁ f₂ i ⟫) i aom (F-idₑ i) = isSet→SquareP (λ i j → 𝓓 .isSetHom) (aom-F idₑ) aom-id - (λ i → arb𝓓 ⟪ F-idₑ i ⟫) - (λ i → semH ⟪ F-idₑ i ⟫) + (λ i → arb𝓓 ⟪ F-idₑ i ⟫) + (λ i → semH ⟪ F-idₑ i ⟫) i aom (F-seqₑ f g i) = isSet→SquareP (λ i j → 𝓓 .isSetHom) (aom-F (f ⋆ₑ g)) (aom-seq (aom-F f) (aom-F g)) - (λ i → arb𝓓 ⟪ F-seqₑ f g i ⟫) - (λ i → semH ⟪ F-seqₑ f g i ⟫) + (λ i → arb𝓓 ⟪ F-seqₑ f g i ⟫) + (λ i → semH ⟪ F-seqₑ f g i ⟫) i aom (isSetFExp f f₁ x y i j) k = isSet→SquareP (λ i j → (isOfHLevelPathP {A = λ k → 𝓓 [ aoo _ k , aoo _ k ]} 2 (𝓓 .isSetHom) - (arb𝓓 ⟪ isSetFExp f f₁ x y i j ⟫) - ((semH ⟪ isSetFExp f f₁ x y i j ⟫)))) + (arb𝓓 ⟪ isSetFExp f f₁ x y i j ⟫) + ((semH ⟪ isSetFExp f f₁ x y i j ⟫)))) (λ j k → aom (x j) k) (λ j k → aom (y j) k) (λ i k → aom f k) diff --git a/Cubical.Categories.Constructions.FullSubcategory.html b/Cubical.Categories.Constructions.FullSubcategory.html index 3c47ff6853..169648966b 100644 --- a/Cubical.Categories.Constructions.FullSubcategory.html +++ b/Cubical.Categories.Constructions.FullSubcategory.html @@ -14,7 +14,7 @@ open import Cubical.Categories.Category open import Cubical.Categories.Isomorphism -open import Cubical.Categories.Functor renaming (𝟙⟨_⟩ to funcId) +open import Cubical.Categories.Functor renaming (𝟙⟨_⟩ to funcId) private variable @@ -24,7 +24,7 @@ private module C = Category C open Category - open Functor + open Functor FullSubcategory : Category (ℓ-max ℓC ℓP) ℓC' ob FullSubcategory = Σ[ x ∈ C.ob ] P x @@ -36,13 +36,13 @@ ⋆Assoc FullSubcategory = C.⋆Assoc isSetHom FullSubcategory = C.isSetHom - FullInclusion : Functor FullSubcategory C - F-ob FullInclusion = fst - F-hom FullInclusion = idfun _ - F-id FullInclusion = refl - F-seq FullInclusion f g = refl + FullInclusion : Functor FullSubcategory C + F-ob FullInclusion = fst + F-hom FullInclusion = idfun _ + F-id FullInclusion = refl + F-seq FullInclusion f g = refl - isFullyFaithfulIncl : isFullyFaithful FullInclusion + isFullyFaithfulIncl : isFullyFaithful FullInclusion isFullyFaithfulIncl _ _ = idEquiv _ .snd module _ (x y : FullSubcategory .ob) where @@ -79,13 +79,13 @@ module C = Category C module D = Category D open Category - open Functor + open Functor - ToFullSubcategory : (F : Functor C D) → ((c : C.ob) → Q (F-ob F c)) → Functor C (FullSubcategory D Q) - F-ob (ToFullSubcategory F f) c = F-ob F c , f c - F-hom (ToFullSubcategory F f) = F-hom F - F-id (ToFullSubcategory F f) = F-id F - F-seq (ToFullSubcategory F f) = F-seq F + ToFullSubcategory : (F : Functor C D) → ((c : C.ob) → Q (F-ob F c)) → Functor C (FullSubcategory D Q) + F-ob (ToFullSubcategory F f) c = F-ob F c , f c + F-hom (ToFullSubcategory F f) = F-hom F + F-id (ToFullSubcategory F f) = F-id F + F-seq (ToFullSubcategory F f) = F-seq F module _ (C : Category ℓC ℓC') (P : Category.ob C → Type ℓP) (D : Category ℓD ℓD') (Q : Category.ob D → Type ℓQ) where @@ -93,23 +93,23 @@ module C = Category C module D = Category D open Category - open Functor + open Functor - MapFullSubcategory : (F : Functor C D) → ((c : C.ob) → P c → Q (F-ob F c)) - → Functor (FullSubcategory C P) (FullSubcategory D Q) + MapFullSubcategory : (F : Functor C D) → ((c : C.ob) → P c → Q (F-ob F c)) + → Functor (FullSubcategory C P) (FullSubcategory D Q) MapFullSubcategory F f = ToFullSubcategory (FullSubcategory C P) D Q - (funcComp F (FullInclusion C P) ) + (funcComp F (FullInclusion C P) ) λ (c , p) → f c p module _ (C : Category ℓC ℓC') (P : Category.ob C → Type ℓP) where private module C = Category C open Category - open Functor + open Functor MapFullSubcategory-id : MapFullSubcategory C P C P (funcId C) (λ c p → p) ≡ funcId (FullSubcategory C P) - MapFullSubcategory-id = Functor≡ + MapFullSubcategory-id = Functor≡ (λ (c , p) → refl) (λ γ → refl) @@ -121,16 +121,16 @@ module D = Category D module E = Category E open Category - open Functor + open Functor MapFullSubcategory-seq : - (F : Functor C D) → (f : (c : C.ob) → P c → Q (F-ob F c)) → - (G : Functor D E) → (g : (d : D.ob) → Q d → R (F-ob G d)) → - MapFullSubcategory C P E R (funcComp G F) (λ c p → g (F-ob F c) (f c p)) ≡ - funcComp + (F : Functor C D) → (f : (c : C.ob) → P c → Q (F-ob F c)) → + (G : Functor D E) → (g : (d : D.ob) → Q d → R (F-ob G d)) → + MapFullSubcategory C P E R (funcComp G F) (λ c p → g (F-ob F c) (f c p)) ≡ + funcComp (MapFullSubcategory D Q E R G g) (MapFullSubcategory C P D Q F f) - MapFullSubcategory-seq F f G g = Functor≡ + MapFullSubcategory-seq F f G g = Functor≡ (λ (c , p) → refl) (λ γ → refl) @@ -144,14 +144,14 @@ {P : C .ob → Type ℓP}(isPropP : (c : C .ob) → isProp (P c)) where - open Functor + open Functor open isUnivalent -- Full subcategory (injective on objects) is injective on objects. - isEmbdIncl-ob : isEmbedding (FullInclusion C P .F-ob) - isEmbdIncl-ob _ _ = isEmbeddingFstΣProp isPropP + isEmbdIncl-ob : isEmbedding (FullInclusion C P .F-ob) + isEmbdIncl-ob _ _ = isEmbeddingFstΣProp isPropP -- Full subcategory (injective on objects) of univalent category is univalent. diff --git a/Cubical.Categories.Constructions.Lift.html b/Cubical.Categories.Constructions.Lift.html index 1491cd056c..ed22f47328 100644 --- a/Cubical.Categories.Constructions.Lift.html +++ b/Cubical.Categories.Constructions.Lift.html @@ -25,15 +25,15 @@ LiftHoms .⋆Assoc f g h = cong lift (C .⋆Assoc (f .lower) (g .lower) (h .lower)) LiftHoms .isSetHom = isOfHLevelLift 2 (C .isSetHom) - liftHoms : Functor C LiftHoms - liftHoms .Functor.F-ob = λ z → z - liftHoms .Functor.F-hom = lift - liftHoms .Functor.F-id = refl - liftHoms .Functor.F-seq f g = refl + liftHoms : Functor C LiftHoms + liftHoms .Functor.F-ob = λ z → z + liftHoms .Functor.F-hom = lift + liftHoms .Functor.F-id = refl + liftHoms .Functor.F-seq f g = refl - lowerHoms : Functor LiftHoms C - lowerHoms .Functor.F-ob = λ z → z - lowerHoms .Functor.F-hom = lower - lowerHoms .Functor.F-id = refl - lowerHoms .Functor.F-seq f g = refl + lowerHoms : Functor LiftHoms C + lowerHoms .Functor.F-ob = λ z → z + lowerHoms .Functor.F-hom = lower + lowerHoms .Functor.F-id = refl + lowerHoms .Functor.F-seq f g = refl \ No newline at end of file diff --git a/Cubical.Categories.Constructions.Power.html b/Cubical.Categories.Constructions.Power.html index c757f79913..8c10bfa3f3 100644 --- a/Cubical.Categories.Constructions.Power.html +++ b/Cubical.Categories.Constructions.Power.html @@ -19,11 +19,11 @@ PowerCategory X C = ΠC X (λ _ → C) PseudoYoneda : {C : Category ℓc ℓc'} - → Functor C (PowerCategory (C .ob) (SET ℓc')) + → Functor C (PowerCategory (C .ob) (SET ℓc')) PseudoYoneda {C = C} = Π-intro _ (λ _ → SET _) λ a → C [ a ,-] isFaithfulPseudoYoneda : {C : Category ℓc ℓc'} - → Functor.isFaithful (PseudoYoneda {C = C}) + → Functor.isFaithful (PseudoYoneda {C = C}) isFaithfulPseudoYoneda {C = C} x y f g p = sym (C .⋆IdL f) ∙ (λ i → p i _ (C .id)) ∙ C .⋆IdL g \ No newline at end of file diff --git a/Cubical.Categories.Constructions.Product.html b/Cubical.Categories.Constructions.Product.html index 5c71a16657..ffdb17f421 100644 --- a/Cubical.Categories.Constructions.Product.html +++ b/Cubical.Categories.Constructions.Product.html @@ -15,7 +15,7 @@ ℓA ℓC ℓC' ℓD ℓD' : Level open Category -open Functor +open Functor module _ (A : Type ℓA) (catC : A → Category ℓC ℓC') where ΠC : Category (ℓ-max ℓA ℓC) (ℓ-max ℓA ℓC') @@ -28,15 +28,15 @@ ΠC .⋆Assoc h g f = funExt λ a → ⋆Assoc (catC a) (h a) (g a) (f a) ΠC .isSetHom = isSetΠ (λ a → isSetHom (catC a)) - Π-intro : {D : Category ℓD ℓD'} → (∀ (a : A) → Functor D (catC a)) → Functor D ΠC - Π-intro Fs .Functor.F-ob d a = Fs a ⟅ d ⟆ - Π-intro Fs .Functor.F-hom f a = Fs a ⟪ f ⟫ - Π-intro Fs .Functor.F-id = funExt (λ a → Fs a .F-id) - Π-intro Fs .Functor.F-seq f g = funExt (λ a → Fs a .F-seq f g) - - πC : (a : A) → Functor ΠC (catC a) - πC a .Functor.F-ob = λ xs → xs a - πC a .Functor.F-hom = λ fs → fs a - πC a .Functor.F-id = refl - πC a .Functor.F-seq fs gs = refl + Π-intro : {D : Category ℓD ℓD'} → (∀ (a : A) → Functor D (catC a)) → Functor D ΠC + Π-intro Fs .Functor.F-ob d a = Fs a ⟅ d ⟆ + Π-intro Fs .Functor.F-hom f a = Fs a ⟪ f ⟫ + Π-intro Fs .Functor.F-id = funExt (λ a → Fs a .F-id) + Π-intro Fs .Functor.F-seq f g = funExt (λ a → Fs a .F-seq f g) + + πC : (a : A) → Functor ΠC (catC a) + πC a .Functor.F-ob = λ xs → xs a + πC a .Functor.F-hom = λ fs → fs a + πC a .Functor.F-id = refl + πC a .Functor.F-seq fs gs = refl \ No newline at end of file diff --git a/Cubical.Categories.Constructions.Quotient.html b/Cubical.Categories.Constructions.Quotient.html index f947554723..c451b6fb56 100644 --- a/Cubical.Categories.Constructions.Quotient.html +++ b/Cubical.Categories.Constructions.Quotient.html @@ -67,12 +67,12 @@ C/~ = QuotientCategory -- Quotient map - open Functor - QuoFunctor : Functor C C/~ - QuoFunctor .F-ob x = x - QuoFunctor .F-hom = ⟦_⟧ - QuoFunctor .F-id = refl - QuoFunctor .F-seq f g = refl + open Functor + QuoFunctor : Functor C C/~ + QuoFunctor .F-ob x = x + QuoFunctor .F-hom = ⟦_⟧ + QuoFunctor .F-id = refl + QuoFunctor .F-seq f g = refl -- Quotients preserve initial / terminal objects isInitial/~ : {z : ob C} → isInitial C z → isInitial C/~ z diff --git a/Cubical.Categories.Constructions.Slice.Properties.html b/Cubical.Categories.Constructions.Slice.Properties.html index 12bbf76c41..712e9ad1a3 100644 --- a/Cubical.Categories.Constructions.Slice.Properties.html +++ b/Cubical.Categories.Constructions.Slice.Properties.html @@ -27,20 +27,20 @@ open Elements.Contravariant {C = C} open _≃ᶜ_ -open Functor +open Functor open WeakInverse -slice→el : Functor (SliceCat C c) (∫ᴾ (C [-, c ])) -slice→el .F-ob s = s .S-ob , s .S-arr -slice→el .F-hom f = f .S-hom , f .S-comm -slice→el .F-id = ΣPathP (refl , (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _)) -slice→el .F-seq _ _ = ΣPathP (refl , (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _)) +slice→el : Functor (SliceCat C c) (∫ᴾ (C [-, c ])) +slice→el .F-ob s = s .S-ob , s .S-arr +slice→el .F-hom f = f .S-hom , f .S-comm +slice→el .F-id = ΣPathP (refl , (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _)) +slice→el .F-seq _ _ = ΣPathP (refl , (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _)) -el→slice : Functor (∫ᴾ (C [-, c ])) (SliceCat C c) -el→slice .F-ob (_ , s) = sliceob s -el→slice .F-hom (f , comm) = slicehom f comm -el→slice .F-id = SliceHom-≡-intro C c refl (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _) -el→slice .F-seq _ _ = SliceHom-≡-intro C c refl (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _) +el→slice : Functor (∫ᴾ (C [-, c ])) (SliceCat C c) +el→slice .F-ob (_ , s) = sliceob s +el→slice .F-hom (f , comm) = slicehom f comm +el→slice .F-id = SliceHom-≡-intro C c refl (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _) +el→slice .F-seq _ _ = SliceHom-≡-intro C c refl (isOfHLevelPath' 1 (C .isSetHom) _ _ _ _) open NatTrans open NatIso diff --git a/Cubical.Categories.Constructions.SubObject.html b/Cubical.Categories.Constructions.SubObject.html index 26911d2b42..a5b3b9e354 100644 --- a/Cubical.Categories.Constructions.SubObject.html +++ b/Cubical.Categories.Constructions.SubObject.html @@ -24,10 +24,10 @@ isSubObj (sliceob arr) = isMonic C arr , isPropIsMonic C arr SubObjCat : Category (ℓ-max ℓ ℓ') ℓ' -SubObjCat = ΣPropCat SliceCat isSubObj +SubObjCat = ΣPropCat SliceCat isSubObj -SubObjCat→SliceCat : Functor SubObjCat SliceCat -SubObjCat→SliceCat = forgetΣPropCat SliceCat isSubObj +SubObjCat→SliceCat : Functor SubObjCat SliceCat +SubObjCat→SliceCat = forgetΣPropCat SliceCat isSubObj subObjMorIsMonic : {s t : SubObjCat .ob} (f : SubObjCat [ s , t ]) → isMonic C (S-hom f) diff --git a/Cubical.Categories.Constructions.TwistedArrow.html b/Cubical.Categories.Constructions.TwistedArrow.html index adce364358..a7d4b918de 100644 --- a/Cubical.Categories.Constructions.TwistedArrow.html +++ b/Cubical.Categories.Constructions.TwistedArrow.html @@ -20,12 +20,12 @@ TwistedArrowCategory : (C : Category ℓ ℓ') → Category (ℓ-max ℓ ℓ') ℓ' TwistedArrowCategory C = ∫ HomFunctor C -TwistedEnds : (C : Category ℓ ℓ') → Functor (TwistedArrowCategory C) (C ^op ×C C) +TwistedEnds : (C : Category ℓ ℓ') → Functor (TwistedArrowCategory C) (C ^op ×C C) TwistedEnds C = ForgetElements (HomFunctor C) -TwistedDom : (C : Category ℓ ℓ') → Functor ((TwistedArrowCategory C) ^op) C -TwistedDom C = ((Fst (C ^op) C) ^opF) ∘F (ForgetElements (HomFunctor C) ^opF) +TwistedDom : (C : Category ℓ ℓ') → Functor ((TwistedArrowCategory C) ^op) C +TwistedDom C = ((Fst (C ^op) C) ^opF) ∘F (ForgetElements (HomFunctor C) ^opF) -TwistedCod : (C : Category ℓ ℓ') → Functor (TwistedArrowCategory C) C -TwistedCod C = (Snd (C ^op) C) ∘F ForgetElements (HomFunctor C) +TwistedCod : (C : Category ℓ ℓ') → Functor (TwistedArrowCategory C) C +TwistedCod C = (Snd (C ^op) C) ∘F ForgetElements (HomFunctor C) \ No newline at end of file diff --git a/Cubical.Categories.Displayed.Adjoint.html b/Cubical.Categories.Displayed.Adjoint.html index b6697389e0..fe891b8505 100644 --- a/Cubical.Categories.Displayed.Adjoint.html +++ b/Cubical.Categories.Displayed.Adjoint.html @@ -21,16 +21,16 @@ -- Adjoint definition 1: unit-counit record _⊣[_]_ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - {F : Functor C D} {G : Functor D C} + {F : Functor C D} {G : Functor D C} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} - (Fᴰ : Functorᴰ F Cᴰ Dᴰ) (A : F UnitCounit.⊣ G) (Gᴰ : Functorᴰ G Dᴰ Cᴰ) + (Fᴰ : Functorᴰ F Cᴰ Dᴰ) (A : F UnitCounit.⊣ G) (Gᴰ : Functorᴰ G Dᴰ Cᴰ) : Type (ℓ-max (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓCᴰ ℓCᴰ')) (ℓ-max (ℓ-max ℓD ℓD') (ℓ-max ℓDᴰ ℓDᴰ'))) where open Category open NatTransᴰ private - module A = UnitCounit._⊣_ A + module A = UnitCounit._⊣_ A module Cᴰ = Categoryᴰ Cᴰ module Dᴰ = Categoryᴰ Dᴰ module Fᴰ = Functorᴰ Fᴰ @@ -38,16 +38,16 @@ field -- unit - ηᴰ : NatTransᴰ A.η 𝟙ᴰ⟨ Cᴰ ⟩ (funcCompᴰ Gᴰ Fᴰ) + ηᴰ : NatTransᴰ A.η 𝟙ᴰ⟨ Cᴰ ⟩ (funcCompᴰ Gᴰ Fᴰ) -- counit - εᴰ : NatTransᴰ A.ε (funcCompᴰ Fᴰ Gᴰ) 𝟙ᴰ⟨ Dᴰ ⟩ + εᴰ : NatTransᴰ A.ε (funcCompᴰ Fᴰ Gᴰ) 𝟙ᴰ⟨ Dᴰ ⟩ -- triangle identities Δ₁ᴰ : {x : C .ob} (xᴰ : Cᴰ.ob[ x ]) → Fᴰ.F-homᴰ (ηᴰ .N-obᴰ xᴰ) Dᴰ.⋆ᴰ εᴰ .N-obᴰ (Fᴰ.F-obᴰ xᴰ) - Dᴰ.≡[ A.Δ₁ x ] + Dᴰ.≡[ A.Δ₁ x ] Dᴰ.idᴰ Δ₂ᴰ : {y : D .ob} (yᴰ : Dᴰ.ob[ y ]) → ηᴰ .N-obᴰ (Gᴰ.F-obᴰ yᴰ) Cᴰ.⋆ᴰ Gᴰ.F-homᴰ (εᴰ .N-obᴰ yᴰ) - Cᴰ.≡[ A.Δ₂ y ] + Cᴰ.≡[ A.Δ₂ y ] Cᴰ.idᴰ \ No newline at end of file diff --git a/Cubical.Categories.Displayed.Cartesian.html b/Cubical.Categories.Displayed.Cartesian.html index 417b85f6ef..b80d9dff54 100644 --- a/Cubical.Categories.Displayed.Cartesian.html +++ b/Cubical.Categories.Displayed.Cartesian.html @@ -96,7 +96,7 @@ -- It trivially composes to the expected morphism. , Homᴰ≡-DiscreteOpfibration _ _) -- And it's trivially equal to any other choice. - , λ _ → Σ≡Prop (λ _ → isSetHomᴰ _ _) (Homᴰ≡-DiscreteOpfibration _ _) + , λ _ → Σ≡Prop (λ _ → isSetHomᴰ _ _) (Homᴰ≡-DiscreteOpfibration _ _) where lift-g : Σ[ cᴰ ∈ ob[ c ] ] Hom[ g ][ bᴰ , cᴰ ] lift-g = unique-lift g bᴰ .fst @@ -120,17 +120,17 @@ uniqueOpcleavageDiscreteOpfibration .snd opcleavage = implicitFunExt $ implicitFunExt $ funExt λ f → funExt λ aᴰ → ΣPathP $ - map-snd (ΣPathPProp isPropIsOpcartesian) $ + map-snd (ΣPathPProp isPropIsOpcartesian) $ PathPΣ $ unique-lift f aᴰ .snd (map-snd fst (opcleavage f aᴰ)) - open Functor - discreteOpfibrationToCopresheaf : Functor B (SET ℓC) - discreteOpfibrationToCopresheaf .F-ob b = ob[ b ] , isSet-fibers b - discreteOpfibrationToCopresheaf .F-hom f a = unique-lift f a .fst .fst - discreteOpfibrationToCopresheaf .F-id = funExt λ a → + open Functor + discreteOpfibrationToCopresheaf : Functor B (SET ℓC) + discreteOpfibrationToCopresheaf .F-ob b = ob[ b ] , isSet-fibers b + discreteOpfibrationToCopresheaf .F-hom f a = unique-lift f a .fst .fst + discreteOpfibrationToCopresheaf .F-id = funExt λ a → cong fst $ unique-lift B.id a .snd (a , idᴰ) - discreteOpfibrationToCopresheaf .F-seq f g = funExt λ a → + discreteOpfibrationToCopresheaf .F-seq f g = funExt λ a → let step1 = unique-lift f a .fst step2 = unique-lift g (step1 .fst) .fst @@ -172,7 +172,7 @@ R.[ _ ]∙[ _ ] Cᴰ.⋆IdLᴰ fgᴰ isIsoᴰ→isOpcartesian g .equiv-proof fgᴰ .snd (gᴰ , gᴰ-infib) = - Σ≡Prop (λ _ → isOfHLevelPathP' 1 Cᴰ.isSetHomᴰ _ _) $ + Σ≡Prop (λ _ → isOfHLevelPathP' 1 Cᴰ.isSetHomᴰ _ _) $ R.≡[]-rectify $ symP (R.reind-filler (basepath g) _) R.[ sym (basepath g) ]∙[ _ ] @@ -225,7 +225,7 @@ (σ : B [ a , b ]) where - open Functor + open Functor open Category -- Nice universal characterization of candidates for arrow substitution. @@ -243,7 +243,7 @@ symP (R.reind-filler _ _)) (cart .fst) , λ g' → - Σ≡Prop + Σ≡Prop (λ _ → isOfHLevelPathP' 1 isSetHomᴰ _ _) (cong fst $ cart .snd $ map-snd @@ -255,10 +255,10 @@ cleavage σ _ .snd .snd _ .equiv-proof $ R.reind (B.⋆IdL _ ∙ sym (B.⋆IdR _)) (f ⋆ᴰ cleavage σ _ .snd .fst) - substitutionFunctor : Functor (VerticalCategory Cᴰ a) (VerticalCategory Cᴰ b) - substitutionFunctor .F-ob c = cleavage σ c .fst - substitutionFunctor .F-hom f = substituteArrow f .fst .fst - substitutionFunctor .F-id {x} = cong fst $ + substitutionFunctor : Functor (VerticalCategory Cᴰ a) (VerticalCategory Cᴰ b) + substitutionFunctor .F-ob c = cleavage σ c .fst + substitutionFunctor .F-hom f = substituteArrow f .fst .fst + substitutionFunctor .F-id {x} = cong fst $ substituteArrow (VerticalCategory Cᴰ a .Category.id) .snd $ VerticalCategory Cᴰ b .Category.id , R.≡[]-rectify @@ -269,7 +269,7 @@ symP (⋆IdLᴰ (cleavage σ x .snd .fst)) R.[ sym (B.⋆IdL σ) ]∙[ _ ] (R.reind-filler refl idᴰ R.[ refl ]⋆[ refl ] refl)) - substitutionFunctor .F-seq {x} {y} {z} f g = cong fst $ + substitutionFunctor .F-seq {x} {y} {z} f g = cong fst $ substituteArrow (VerticalCategory Cᴰ a .Category._⋆_ f g) .snd $ VerticalCategory Cᴰ b .Category._⋆_ (stepf .fst) (stepg .fst) , R.≡[]-rectify diff --git a/Cubical.Categories.Displayed.Functor.html b/Cubical.Categories.Displayed.Functor.html index 47de885258..30d575f1a8 100644 --- a/Cubical.Categories.Displayed.Functor.html +++ b/Cubical.Categories.Displayed.Functor.html @@ -20,34 +20,34 @@ ℓB ℓB' ℓBᴰ ℓBᴰ' ℓC ℓC' ℓCᴰ ℓCᴰ' ℓD ℓD' ℓDᴰ ℓDᴰ' ℓE ℓE' ℓEᴰ ℓEᴰ' : Level record Functorᴰ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - (F : Functor C D) (Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ') (Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ') + (F : Functor C D) (Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ') (Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ') : Type (ℓ-max (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) (ℓ-max (ℓ-max ℓCᴰ ℓCᴰ') (ℓ-max ℓDᴰ ℓDᴰ'))) where no-eta-equality open Category - open Functor F + open Functor F private module Cᴰ = Categoryᴰ Cᴰ module Dᴰ = Categoryᴰ Dᴰ field - F-obᴰ : {x : C .ob} → Cᴰ.ob[ x ] → Dᴰ.ob[ F-ob x ] + F-obᴰ : {x : C .ob} → Cᴰ.ob[ x ] → Dᴰ.ob[ F-ob x ] F-homᴰ : {x y : C .ob} {f : C [ x , y ]} {xᴰ : Cᴰ.ob[ x ]} {yᴰ : Cᴰ.ob[ y ]} - → Cᴰ [ f ][ xᴰ , yᴰ ] → Dᴰ [ F-hom f ][ F-obᴰ xᴰ , F-obᴰ yᴰ ] + → Cᴰ [ f ][ xᴰ , yᴰ ] → Dᴰ [ F-hom f ][ F-obᴰ xᴰ , F-obᴰ yᴰ ] F-idᴰ : {x : C .ob} {xᴰ : Cᴰ.ob[ x ]} - → F-homᴰ (Cᴰ.idᴰ {p = xᴰ}) Dᴰ.≡[ F-id {x} ] (Dᴰ.idᴰ {p = F-obᴰ xᴰ}) + → F-homᴰ (Cᴰ.idᴰ {p = xᴰ}) Dᴰ.≡[ F-id {x} ] (Dᴰ.idᴰ {p = F-obᴰ xᴰ}) F-seqᴰ : {x y z : C .ob} {f : C [ x , y ]} {g : C [ y , z ]} {xᴰ : Cᴰ.ob[ x ]} {yᴰ : Cᴰ.ob[ y ]} {zᴰ : Cᴰ.ob[ z ]} (fᴰ : Cᴰ [ f ][ xᴰ , yᴰ ]) (gᴰ : Cᴰ [ g ][ yᴰ , zᴰ ]) - → F-homᴰ (fᴰ Cᴰ.⋆ᴰ gᴰ) Dᴰ.≡[ F-seq f g ] F-homᴰ fᴰ Dᴰ.⋆ᴰ F-homᴰ gᴰ + → F-homᴰ (fᴰ Cᴰ.⋆ᴰ gᴰ) Dᴰ.≡[ F-seq f g ] F-homᴰ fᴰ Dᴰ.⋆ᴰ F-homᴰ gᴰ -module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} {F G : Functor C D} {H : F ≡ G} +module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} {F G : Functor C D} {H : F ≡ G} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} {Fᴰ : Functorᴰ F Cᴰ Dᴰ} {Gᴰ : Functorᴰ G Cᴰ Dᴰ} where open Category - open Functor + open Functor open Functorᴰ private module Cᴰ = Categoryᴰ Cᴰ @@ -56,15 +56,15 @@ module Gᴰ = Functorᴰ Gᴰ Functorᴰ≡ : - (Hᴰ : {x : C .ob} (xᴰ : Cᴰ.ob[ x ]) → PathP (λ i → Dᴰ.ob[ H i ⟅ x ⟆ ]) (Fᴰ.F-obᴰ xᴰ) (Gᴰ.F-obᴰ xᴰ)) + (Hᴰ : {x : C .ob} (xᴰ : Cᴰ.ob[ x ]) → PathP (λ i → Dᴰ.ob[ H i ⟅ x ⟆ ]) (Fᴰ.F-obᴰ xᴰ) (Gᴰ.F-obᴰ xᴰ)) → ({x y : C .ob} {f : C [ x , y ]} {xᴰ : Cᴰ.ob[ x ]} {yᴰ : Cᴰ.ob[ y ]} (fᴰ : Cᴰ [ f ][ xᴰ , yᴰ ]) - → PathP (λ i → Dᴰ [ H i ⟪ f ⟫ ][ Hᴰ xᴰ i , Hᴰ yᴰ i ]) (Fᴰ.F-homᴰ fᴰ) (Gᴰ.F-homᴰ fᴰ)) + → PathP (λ i → Dᴰ [ H i ⟪ f ⟫ ][ Hᴰ xᴰ i , Hᴰ yᴰ i ]) (Fᴰ.F-homᴰ fᴰ) (Gᴰ.F-homᴰ fᴰ)) → PathP (λ i → Functorᴰ (H i) Cᴰ Dᴰ) Fᴰ Gᴰ Functorᴰ≡ Hᴰ α i .F-obᴰ xᴰ = Hᴰ xᴰ i Functorᴰ≡ Hᴰ α i .F-homᴰ fᴰ = α fᴰ i Functorᴰ≡ Hᴰ α i .F-idᴰ {xᴰ = xᴰ} = isProp→PathP - {B = λ i → PathP (λ j → Dᴰ [ H i .F-id j ][ Hᴰ xᴰ i , Hᴰ xᴰ i ]) (α (Cᴰ.idᴰ) i) Dᴰ.idᴰ} + {B = λ i → PathP (λ j → Dᴰ [ H i .F-id j ][ Hᴰ xᴰ i , Hᴰ xᴰ i ]) (α (Cᴰ.idᴰ) i) Dᴰ.idᴰ} (λ i → isOfHLevelPathP' 1 Dᴰ.isSetHomᴰ _ _) Fᴰ.F-idᴰ Gᴰ.F-idᴰ @@ -72,7 +72,7 @@ Functorᴰ≡ Hᴰ α i .F-seqᴰ {f = f} {g = g} {xᴰ = xᴰ} {zᴰ = zᴰ} fᴰ gᴰ = isProp→PathP {B = λ i → - PathP (λ j → Dᴰ [ H i .F-seq f g j ][ Hᴰ xᴰ i , Hᴰ zᴰ i ]) + PathP (λ j → Dᴰ [ H i .F-seq f g j ][ Hᴰ xᴰ i , Hᴰ zᴰ i ]) (α (fᴰ Cᴰ.⋆ᴰ gᴰ) i) (α fᴰ i Dᴰ.⋆ᴰ α gᴰ i)} (λ i → isOfHLevelPathP' 1 Dᴰ.isSetHomᴰ _ _) (Fᴰ.F-seqᴰ fᴰ gᴰ) @@ -84,7 +84,7 @@ module _ where open Functorᴰ - 𝟙ᴰ⟨_⟩ : {C : Category ℓC ℓC'} (Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ') → Functorᴰ 𝟙⟨ C ⟩ Cᴰ Cᴰ + 𝟙ᴰ⟨_⟩ : {C : Category ℓC ℓC'} (Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ') → Functorᴰ 𝟙⟨ C ⟩ Cᴰ Cᴰ 𝟙ᴰ⟨ Cᴰ ⟩ .F-obᴰ x = x 𝟙ᴰ⟨ Cᴰ ⟩ .F-homᴰ f = f 𝟙ᴰ⟨ Cᴰ ⟩ .F-idᴰ = refl @@ -93,7 +93,7 @@ -- Displayed functor composition module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} {E : Category ℓE ℓE'} - {F : Functor C D} {G : Functor D E} + {F : Functor C D} {G : Functor D E} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} {Eᴰ : Categoryᴰ E ℓEᴰ ℓEᴰ'} (Gᴰ : Functorᴰ G Dᴰ Eᴰ) (Fᴰ : Functorᴰ F Cᴰ Dᴰ) where @@ -103,7 +103,7 @@ module Fᴰ = Functorᴰ Fᴰ module Gᴰ = Functorᴰ Gᴰ - funcCompᴰ : Functorᴰ (funcComp G F) Cᴰ Eᴰ + funcCompᴰ : Functorᴰ (funcComp G F) Cᴰ Eᴰ funcCompᴰ .F-obᴰ = Gᴰ.F-obᴰ ∘ Fᴰ.F-obᴰ funcCompᴰ .F-homᴰ = Gᴰ.F-homᴰ ∘ Fᴰ.F-homᴰ funcCompᴰ .F-idᴰ = compPathP' {B = Eᴰ [_][ _ , _ ]} (congP (λ _ → Gᴰ.F-homᴰ) Fᴰ.F-idᴰ) Gᴰ.F-idᴰ @@ -115,7 +115,7 @@ -- Displayed functor associativity F-assocᴰ : {B : Category ℓB ℓB'} {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} {E : Category ℓE ℓE'} - {F : Functor B C} {G : Functor C D} {H : Functor D E} + {F : Functor B C} {G : Functor C D} {H : Functor D E} {Bᴰ : Categoryᴰ B ℓBᴰ ℓBᴰ'} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} {Eᴰ : Categoryᴰ E ℓEᴰ ℓEᴰ'} {Fᴰ : Functorᴰ F Bᴰ Cᴰ} {Gᴰ : Functorᴰ G Cᴰ Dᴰ} {Hᴰ : Functorᴰ H Dᴰ Eᴰ} @@ -124,7 +124,7 @@ -- Displayed functor unit laws -module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} {F : Functor C D} +module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} {F : Functor C D} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} {Fᴰ : Functorᴰ F Cᴰ Dᴰ} where open Functorᴰ @@ -145,17 +145,17 @@ -- Total functor of a displayed functor module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - {F : Functor C D} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} + {F : Functor C D} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} (Fᴰ : Functorᴰ F Cᴰ Dᴰ) where - open Functor + open Functor private module Fᴰ = Functorᴰ Fᴰ - ∫F : Functor (∫C Cᴰ) (∫C Dᴰ) - ∫F .F-ob (x , xᴰ) = _ , Fᴰ.F-obᴰ xᴰ - ∫F .F-hom (_ , fᴰ) = _ , Fᴰ.F-homᴰ fᴰ - ∫F .F-id = ΣPathP (_ , Fᴰ.F-idᴰ) - ∫F .F-seq _ _ = ΣPathP (_ , (Fᴰ.F-seqᴰ _ _)) + ∫F : Functor (∫C Cᴰ) (∫C Dᴰ) + ∫F .F-ob (x , xᴰ) = _ , Fᴰ.F-obᴰ xᴰ + ∫F .F-hom (_ , fᴰ) = _ , Fᴰ.F-homᴰ fᴰ + ∫F .F-id = ΣPathP (_ , Fᴰ.F-idᴰ) + ∫F .F-seq _ _ = ΣPathP (_ , (Fᴰ.F-seqᴰ _ _)) \ No newline at end of file diff --git a/Cubical.Categories.Displayed.Instances.Codomain.html b/Cubical.Categories.Displayed.Instances.Codomain.html index 161c1a9cd1..fb4ca2c2de 100644 --- a/Cubical.Categories.Displayed.Instances.Codomain.html +++ b/Cubical.Categories.Displayed.Instances.Codomain.html @@ -34,10 +34,10 @@ ∙ sym (C.⋆Assoc _ _ _) ∙ cong (C._⋆ _) fᴰ-comm ∙ C.⋆Assoc _ _ _ - codomainᴰ .⋆IdLᴰ (fᴰ , _) = ΣPathPProp (λ _ → C.isSetHom _ _) (C.⋆IdL fᴰ) - codomainᴰ .⋆IdRᴰ (fᴰ , _) = ΣPathPProp (λ _ → C.isSetHom _ _) (C.⋆IdR fᴰ) + codomainᴰ .⋆IdLᴰ (fᴰ , _) = ΣPathPProp (λ _ → C.isSetHom _ _) (C.⋆IdL fᴰ) + codomainᴰ .⋆IdRᴰ (fᴰ , _) = ΣPathPProp (λ _ → C.isSetHom _ _) (C.⋆IdR fᴰ) codomainᴰ .⋆Assocᴰ (fᴰ , _) (gᴰ , _) (hᴰ , _) = - ΣPathPProp (λ _ → C.isSetHom _ _) (C.⋆Assoc fᴰ gᴰ hᴰ) + ΣPathPProp (λ _ → C.isSetHom _ _) (C.⋆Assoc fᴰ gᴰ hᴰ) codomainᴰ .isSetHomᴰ = isSetΣSndProp C.isSetHom (λ _ → C.isSetHom _ _) open Covariant @@ -47,9 +47,9 @@ codomainOpcleavage _ _ .snd .fst = C.id , C.⋆IdL _ codomainOpcleavage _ _ .snd .snd _ .equiv-proof (gᴰ , gᴰ-commutes) .fst = (gᴰ , gᴰ-commutes ∙ sym (C.⋆Assoc _ _ _)) - , Σ≡Prop (λ _ → C.isSetHom _ _) (C.⋆IdL _) + , Σ≡Prop (λ _ → C.isSetHom _ _) (C.⋆IdL _) codomainOpcleavage _ _ .snd .snd _ .equiv-proof _ .snd (_ , infib) = - Σ≡Prop (λ _ → isSetΣSndProp C.isSetHom (λ _ → C.isSetHom _ _) _ _) $ - Σ≡Prop (λ _ → C.isSetHom _ _) $ + Σ≡Prop (λ _ → isSetΣSndProp C.isSetHom (λ _ → C.isSetHom _ _) _ _) $ + Σ≡Prop (λ _ → C.isSetHom _ _) $ sym (cong fst infib) ∙ C.⋆IdL _ \ No newline at end of file diff --git a/Cubical.Categories.Displayed.NaturalTransformation.html b/Cubical.Categories.Displayed.NaturalTransformation.html index 7c46582b53..62d7086709 100644 --- a/Cubical.Categories.Displayed.NaturalTransformation.html +++ b/Cubical.Categories.Displayed.NaturalTransformation.html @@ -19,7 +19,7 @@ module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} where record NatTransᴰ - {F : Functor C D} {G : Functor C D} + {F : Functor C D} {G : Functor C D} (α : NatTrans F G) {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} (Fᴰ : Functorᴰ F Cᴰ Dᴰ) (Gᴰ : Functorᴰ G Cᴰ Dᴰ) @@ -44,7 +44,7 @@ open NatTrans open NatTransᴰ - module _ {F G H : Functor C D} {α : NatTrans F G} {β : NatTrans G H} + module _ {F G H : Functor C D} {α : NatTrans F G} {β : NatTrans G H} {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} {Dᴰ : Categoryᴰ D ℓDᴰ ℓDᴰ'} {Fᴰ : Functorᴰ F Cᴰ Dᴰ} {Gᴰ : Functorᴰ G Cᴰ Dᴰ} {Hᴰ : Functorᴰ H Cᴰ Dᴰ} where diff --git a/Cubical.Categories.Displayed.Properties.html b/Cubical.Categories.Displayed.Properties.html index 92238a96d5..002fece041 100644 --- a/Cubical.Categories.Displayed.Properties.html +++ b/Cubical.Categories.Displayed.Properties.html @@ -20,7 +20,7 @@ module _ {C : Category ℓC ℓ'C} {D : Category ℓD ℓ'D} - (Dᴰ : Categoryᴰ D ℓDᴰ ℓ'Dᴰ) (F : Functor C D) + (Dᴰ : Categoryᴰ D ℓDᴰ ℓ'Dᴰ) (F : Functor C D) where private @@ -29,35 +29,35 @@ module D = Category D open Categoryᴰ Dᴰ - open Functor F + open Functor F reindex : Categoryᴰ C ℓDᴰ ℓ'Dᴰ - reindex .Categoryᴰ.ob[_] c = ob[ F-ob c ] - reindex .Categoryᴰ.Hom[_][_,_] f aᴰ bᴰ = Hom[ F-hom f ][ aᴰ , bᴰ ] - reindex .Categoryᴰ.idᴰ = R.reind (sym F-id) idᴰ - reindex .Categoryᴰ._⋆ᴰ_ fᴰ gᴰ = R.reind (sym $ F-seq _ _) (fᴰ ⋆ᴰ gᴰ) + reindex .Categoryᴰ.ob[_] c = ob[ F-ob c ] + reindex .Categoryᴰ.Hom[_][_,_] f aᴰ bᴰ = Hom[ F-hom f ][ aᴰ , bᴰ ] + reindex .Categoryᴰ.idᴰ = R.reind (sym F-id) idᴰ + reindex .Categoryᴰ._⋆ᴰ_ fᴰ gᴰ = R.reind (sym $ F-seq _ _) (fᴰ ⋆ᴰ gᴰ) reindex .Categoryᴰ.⋆IdLᴰ fᴰ = R.≡[]-rectify $ - R.reind-filler-sym (F-seq _ _) _ + R.reind-filler-sym (F-seq _ _) _ R.[ _ ]∙[ _ ] - (R.reind-filler-sym F-id idᴰ R.[ _ ]⋆[ refl ] refl) + (R.reind-filler-sym F-id idᴰ R.[ _ ]⋆[ refl ] refl) R.[ _ ]∙[ _ ] ⋆IdLᴰ fᴰ reindex .Categoryᴰ.⋆IdRᴰ fᴰ = R.≡[]-rectify $ - R.reind-filler-sym (F-seq _ _) _ + R.reind-filler-sym (F-seq _ _) _ R.[ _ ]∙[ _ ] - (refl R.[ refl ]⋆[ _ ] R.reind-filler-sym F-id idᴰ) + (refl R.[ refl ]⋆[ _ ] R.reind-filler-sym F-id idᴰ) R.[ _ ]∙[ _ ] ⋆IdRᴰ fᴰ reindex .Categoryᴰ.⋆Assocᴰ fᴰ gᴰ hᴰ = R.≡[]-rectify $ - R.reind-filler-sym (F-seq _ _) _ + R.reind-filler-sym (F-seq _ _) _ R.[ _ ]∙[ _ ] - (R.reind-filler-sym (F-seq _ _) _ R.[ _ ]⋆[ refl ] refl) + (R.reind-filler-sym (F-seq _ _) _ R.[ _ ]⋆[ refl ] refl) R.[ _ ]∙[ _ ] ⋆Assocᴰ fᴰ gᴰ hᴰ R.[ _ ]∙[ _ ] - (refl R.[ refl ]⋆[ _ ] R.reind-filler (sym $ F-seq _ _) _) + (refl R.[ refl ]⋆[ _ ] R.reind-filler (sym $ F-seq _ _) _) R.[ _ ]∙[ _ ] - R.reind-filler (sym $ F-seq _ _) _ + R.reind-filler (sym $ F-seq _ _) _ reindex .Categoryᴰ.isSetHomᴰ = isSetHomᴰ diff --git a/Cubical.Categories.DistLatticeSheaf.Base.html b/Cubical.Categories.DistLatticeSheaf.Base.html index 30797c1cb5..96ec9676b5 100644 --- a/Cubical.Categories.DistLatticeSheaf.Base.html +++ b/Cubical.Categories.DistLatticeSheaf.Base.html @@ -40,7 +40,7 @@ module _ (L : DistLattice ℓ) (C : Category ℓ' ℓ'') where open Category hiding (_⋆_ ; _∘_) - open Functor + open Functor open Order (DistLattice→Lattice L) open DistLatticeStr (snd L) open JoinSemilattice (Lattice→JoinSemilattice (DistLattice→Lattice L)) @@ -54,7 +54,7 @@ -- C-valued presheaves on a distributive lattice DLPreSheaf : Type (ℓ-max (ℓ-max ℓ ℓ') ℓ'') - DLPreSheaf = Functor (DLCat ^op) C + DLPreSheaf = Functor (DLCat ^op) C module _ (x y : L .fst)where hom-∨₁ : DLCat [ x , x ∨l y ] @@ -88,12 +88,12 @@ F(y) ------→ F(x ∧ y) -} Fsq : (F : DLPreSheaf) - → F .F-hom hom-∨₂ ⋆⟨ C ⟩ F .F-hom hom-∧₂ ≡ - F .F-hom hom-∨₁ ⋆⟨ C ⟩ F .F-hom hom-∧₁ - Fsq F = F-square F sq + → F .F-hom hom-∨₂ ⋆⟨ C ⟩ F .F-hom hom-∧₂ ≡ + F .F-hom hom-∨₁ ⋆⟨ C ⟩ F .F-hom hom-∧₁ + Fsq F = F-square F sq isDLSheafPullback : (F : DLPreSheaf) → Type (ℓ-max (ℓ-max ℓ ℓ') ℓ'') - isDLSheafPullback F = isTerminal C (F-ob F 0l) + isDLSheafPullback F = isTerminal C (F-ob F 0l) × ((x y : L .fst) → isPullback C _ _ _ (Fsq x y F)) isPropIsDLSheafPullback : ∀ F → isProp (isDLSheafPullback F) @@ -114,7 +114,7 @@ open LimCone isDLSheafLimCone : (F : DLPreSheaf) → isDLSheaf F - → (n : ℕ) (α : FinVec (fst L) n) → LimCone (F ∘F (FinVec→Diag L α)) + → (n : ℕ) (α : FinVec (fst L) n) → LimCone (F ∘F (FinVec→Diag L α)) lim (isDLSheafLimCone F isSheafF n α) = _ limCone (isDLSheafLimCone F isSheafF n α) = F-cone F (⋁Cone L α) univProp (isDLSheafLimCone F isSheafF n α) = isSheafF n α @@ -140,27 +140,27 @@ F≤PathPLemma : ∀ {x y z w : ob DLCat} (p : x ≡ y) (q : z ≡ w) (x≥z : (DLCat ^op) [ x , z ]) (y≥w : (DLCat ^op) [ y , w ]) - → PathP (λ i → C [ F .F-ob (p i) , F .F-ob (q i) ]) (F .F-hom x≥z) (F .F-hom y≥w) - F≤PathPLemma p q x≥z y≥w i = F .F-hom (≤PathPLemma p q x≥z y≥w i) + → PathP (λ i → C [ F .F-ob (p i) , F .F-ob (q i) ]) (F .F-hom x≥z) (F .F-hom y≥w) + F≤PathPLemma p q x≥z y≥w i = F .F-hom (≤PathPLemma p q x≥z y≥w i) L→P : isDLSheaf F → isDLSheafPullback F fst (L→P isSheafF) = isTerminalF0 where -- F(0) ≡ terminal obj. - isLimConeF0 : isLimCone _ (F .F-ob 0l) (F-cone F (⋁Cone L (λ ()))) + isLimConeF0 : isLimCone _ (F .F-ob 0l) (F-cone F (⋁Cone L (λ ()))) isLimConeF0 = isSheafF 0 (λ ()) - toCone : (y : ob C) → Cone (funcComp F (FinVec→Diag L {n = 0} (λ ()))) y + toCone : (y : ob C) → Cone (funcComp F (FinVec→Diag L {n = 0} (λ ()))) y coneOut (toCone y) (sing ()) coneOut (toCone y) (pair () _ _) coneOutCommutes (toCone y) {u = sing ()} idAr coneOutCommutes (toCone y) {u = pair () _ _} idAr - toConeMor : ∀ (y : ob C) (f : C [ y , F .F-ob 0l ]) + toConeMor : ∀ (y : ob C) (f : C [ y , F .F-ob 0l ]) → isConeMor (toCone y) (F-cone F (⋁Cone L (λ ()))) f toConeMor y f (sing ()) toConeMor y f (pair () _ _) - isTerminalF0 : isTerminal C (F .F-ob 0l) + isTerminalF0 : isTerminal C (F .F-ob 0l) fst (isTerminalF0 y) = isLimConeF0 _ (toCone y) .fst .fst snd (isTerminalF0 y) f = cong fst (isLimConeF0 _ (toCone y) .snd (_ , toConeMor y f)) @@ -170,22 +170,22 @@ xyVec zero = y xyVec one = x - inducedLimCone : LimCone (funcComp F (FinVec→Diag L xyVec)) - lim inducedLimCone = F .F-ob (⋁ xyVec) + inducedLimCone : LimCone (funcComp F (FinVec→Diag L xyVec)) + lim inducedLimCone = F .F-ob (⋁ xyVec) limCone inducedLimCone = F-cone F (⋁Cone L xyVec) univProp inducedLimCone = isSheafF 2 xyVec - theCone : Cone (funcComp F (FinVec→Diag L xyVec)) (F .F-ob (x ∨l y)) - coneOut (theCone) (sing zero) = F .F-hom (hom-∨₂ x y) - coneOut (theCone) (sing one) = F .F-hom (hom-∨₁ x y) + theCone : Cone (funcComp F (FinVec→Diag L xyVec)) (F .F-ob (x ∨l y)) + coneOut (theCone) (sing zero) = F .F-hom (hom-∨₂ x y) + coneOut (theCone) (sing one) = F .F-hom (hom-∨₁ x y) coneOut (theCone) (pair zero zero ()) coneOut (theCone) (pair zero one (s≤s z≤)) = - F .F-hom (hom-∨₂ x y) ⋆⟨ C ⟩ F .F-hom (hom-∧₂ x y) + F .F-hom (hom-∨₂ x y) ⋆⟨ C ⟩ F .F-hom (hom-∧₂ x y) coneOut (theCone) (pair one zero ()) coneOut (theCone) (pair one one (s≤s ())) coneOut (theCone) (pair (suc (suc _)) one (s≤s ())) coneOutCommutes (theCone) {u = u} idAr = - cong (seq' C (coneOut (theCone) u)) (F .F-id) ∙ ⋆IdR C _ + cong (seq' C (coneOut (theCone) u)) (F .F-id) ∙ ⋆IdR C _ coneOutCommutes (theCone) (singPairL {zero} {zero} {()}) coneOutCommutes (theCone) (singPairL {zero} {one} {s≤s z≤}) = refl coneOutCommutes (theCone) (singPairL {one} {zero} {()}) @@ -197,7 +197,7 @@ coneOutCommutes (theCone) (singPairR {one} {one} {s≤s ()}) coneOutCommutes (theCone) (singPairR {suc (suc _)} {one} {s≤s ()}) - theLimCone : LimCone (funcComp F (FinVec→Diag L xyVec)) + theLimCone : LimCone (funcComp F (FinVec→Diag L xyVec)) lim theLimCone = _ limCone theLimCone = theCone univProp theLimCone = @@ -207,26 +207,26 @@ xyPath = cong (y ∨l_) (∨lRid x) ∙ ∨lComm _ _ limPath : lim inducedLimCone ≡ lim theLimCone - limPath = cong (F .F-ob) xyPath + limPath = cong (F .F-ob) xyPath - limConePathP : PathP (λ i → Cone (funcComp F (FinVec→Diag L xyVec)) (limPath i)) + limConePathP : PathP (λ i → Cone (funcComp F (FinVec→Diag L xyVec)) (limPath i)) (limCone inducedLimCone) theCone limConePathP = conePathPOb helperPathP where helperPathP : - ∀ v → PathP (λ i → C [ limPath i , F-ob (funcComp F (FinVec→Diag L xyVec)) v ]) + ∀ v → PathP (λ i → C [ limPath i , F-ob (funcComp F (FinVec→Diag L xyVec)) v ]) (coneOut (limCone inducedLimCone) v) (coneOut theCone v) helperPathP (sing zero) = F≤PathPLemma xyPath refl (ind≤⋁ xyVec zero) (hom-∨₂ x y) helperPathP (sing one) = F≤PathPLemma xyPath refl (ind≤⋁ xyVec one) (hom-∨₁ x y) helperPathP (pair zero zero ()) helperPathP (pair zero one (s≤s z≤)) = - subst (λ f → PathP (λ i → C [ limPath i , F .F-ob (x ∧l y) ]) + subst (λ f → PathP (λ i → C [ limPath i , F .F-ob (x ∧l y) ]) (coneOut (limCone inducedLimCone) (pair zero one (s≤s z≤))) f) - (F-seq F _ _) helperHelperPathP + (F-seq F _ _) helperHelperPathP where - helperHelperPathP : PathP (λ i → C [ limPath i , F .F-ob (x ∧l y) ]) + helperHelperPathP : PathP (λ i → C [ limPath i , F .F-ob (x ∧l y) ]) (coneOut (limCone inducedLimCone) (pair zero one (s≤s z≤))) - (F .F-hom (hom-∨₂ x y ⋆⟨ (DLCat ^op) ⟩ hom-∧₂ x y)) + (F .F-hom (hom-∨₂ x y ⋆⟨ (DLCat ^op) ⟩ hom-∧₂ x y)) helperHelperPathP = F≤PathPLemma xyPath refl (is-trans _ (xyVec zero) _ (≤m→≤j _ _ (∧≤LCancel _ _)) (ind≤⋁ xyVec zero)) (hom-∨₂ x y ⋆⟨ (DLCat ^op) ⟩ hom-∧₂ x y) @@ -242,7 +242,7 @@ snd (fst (P→L (isTerminalF0 , _) ℕ.zero α c cc)) (sing ()) snd (fst (P→L (isTerminalF0 , _) ℕ.zero α c cc)) (pair () _ _) snd (P→L (isTerminalF0 , _) ℕ.zero α c cc) _ = - Σ≡Prop (isPropIsConeMor _ _) (isTerminalF0 c .snd _) + Σ≡Prop (isPropIsConeMor _ _) (isTerminalF0 c .snd _) P→L (F0=1 , presPBSq) (ℕ.suc n) α c cc = uniqueExists (uniqH .fst .fst) @@ -258,20 +258,20 @@ -- the square we want theCospan : Cospan C - l theCospan = F .F-ob (⋁ (α ∘ suc)) - m theCospan = F .F-ob (⋁ β) - r theCospan = F .F-ob (α zero) - s₁ theCospan = F .F-hom (≤-⋁Ext _ _ λ _ → hom-∧₁ _ _) - s₂ theCospan = F .F-hom (⋁IsMax _ _ λ _ → hom-∧₂ _ _) + l theCospan = F .F-ob (⋁ (α ∘ suc)) + m theCospan = F .F-ob (⋁ β) + r theCospan = F .F-ob (α zero) + s₁ theCospan = F .F-hom (≤-⋁Ext _ _ λ _ → hom-∧₁ _ _) + s₂ theCospan = F .F-hom (⋁IsMax _ _ λ _ → hom-∧₂ _ _) thePB : Pullback C theCospan - pbOb thePB = F .F-ob (⋁ α) - pbPr₁ thePB = F .F-hom (hom-∨₂ _ _) - pbPr₂ thePB = F .F-hom (hom-∨₁ _ _) - pbCommutes thePB = F-square F (is-prop-valued _ _ _ _) + pbOb thePB = F .F-ob (⋁ α) + pbPr₁ thePB = F .F-hom (hom-∨₂ _ _) + pbPr₂ thePB = F .F-hom (hom-∨₁ _ _) + pbCommutes thePB = F-square F (is-prop-valued _ _ _ _) univProp thePB f g square = presPBSq (α zero) (⋁ (α ∘ suc)) f g squarePath where - squarePath : f ⋆⟨ C ⟩ F .F-hom (hom-∧₂ _ _) ≡ g ⋆⟨ C ⟩ F .F-hom (hom-∧₁ _ _) + squarePath : f ⋆⟨ C ⟩ F .F-hom (hom-∧₂ _ _) ≡ g ⋆⟨ C ⟩ F .F-hom (hom-∧₁ _ _) squarePath = transport (λ i → f ⋆⟨ C ⟩ F≤PathPLemma refl αβPath (hom-∧₂ _ _) (≤-⋁Ext _ _ λ _ → hom-∧₁ _ _) (~ i) @@ -280,7 +280,7 @@ square -- the two induced cones on which we use the ind. hyp. - ccSuc : Cone (funcComp F (FinVec→Diag L (α ∘ suc))) c + ccSuc : Cone (funcComp F (FinVec→Diag L (α ∘ suc))) c coneOut ccSuc (sing i) = coneOut cc (sing (suc i)) coneOut ccSuc (pair i j i<j) = coneOut cc (pair (suc i) (suc j) (s≤s i<j)) coneOutCommutes ccSuc {sing _} idAr = coneOutCommutes cc idAr @@ -288,83 +288,83 @@ coneOutCommutes ccSuc singPairL = coneOutCommutes cc singPairL coneOutCommutes ccSuc singPairR = coneOutCommutes cc singPairR - cc∧Suc : Cone (funcComp F (FinVec→Diag L β)) c + cc∧Suc : Cone (funcComp F (FinVec→Diag L β)) c coneOut cc∧Suc (sing i) = coneOut cc (pair zero (suc i) (s≤s z≤)) coneOut cc∧Suc (pair i j i<j) = coneOut cc (pair (suc i) (suc j) (s≤s i<j)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) --(αⱼ ∧ αᵢ) ≥ (αⱼ ∧ α₀) ∧ (αᵢ ∧ α₀) coneOutCommutes cc∧Suc idAr = - cong (seq' C (coneOut cc∧Suc _)) ((funcComp F (FinVec→Diag L β)) .F-id) ∙ ⋆IdR C _ + cong (seq' C (coneOut cc∧Suc _)) ((funcComp F (FinVec→Diag L β)) .F-id) ∙ ⋆IdR C _ coneOutCommutes cc∧Suc (singPairL {i = i} {j = j} {i<j = i<j}) = coneOut cc (pair zero (suc i) (s≤s z≤)) - ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairL) - ≡⟨ cong (λ x → seq' C x (funcComp F (FinVec→Diag L β) .F-hom singPairL)) + ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairL) + ≡⟨ cong (λ x → seq' C x (funcComp F (FinVec→Diag L β) .F-hom singPairL)) (sym (coneOutCommutes cc singPairR)) ⟩ - (coneOut cc (sing (suc i)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairR)) - ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairL) + (coneOut cc (sing (suc i)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairR)) + ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairL) ≡⟨ ⋆Assoc C _ _ _ ⟩ - coneOut cc (sing (suc i)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairR) - ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairL)) - ≡⟨ cong (λ x → coneOut cc (sing (suc i)) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - coneOut cc (sing (suc i)) ⋆⟨ C ⟩ F .F-hom - ((FinVec→Diag L α) .F-hom (singPairR {i<j = s≤s z≤}) - ⋆⟨ DLCat ^op ⟩ (FinVec→Diag L β) .F-hom (singPairL {i<j = i<j})) - ≡⟨ cong (λ x → coneOut cc (sing (suc i)) ⋆⟨ C ⟩ F .F-hom x) + coneOut cc (sing (suc i)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairR) + ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairL)) + ≡⟨ cong (λ x → coneOut cc (sing (suc i)) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + coneOut cc (sing (suc i)) ⋆⟨ C ⟩ F .F-hom + ((FinVec→Diag L α) .F-hom (singPairR {i<j = s≤s z≤}) + ⋆⟨ DLCat ^op ⟩ (FinVec→Diag L β) .F-hom (singPairL {i<j = i<j})) + ≡⟨ cong (λ x → coneOut cc (sing (suc i)) ⋆⟨ C ⟩ F .F-hom x) (is-prop-valued _ _ _ _) ⟩ - coneOut cc (sing (suc i)) ⋆⟨ C ⟩ F .F-hom - ((FinVec→Diag L α) .F-hom (singPairL {i<j = s≤s i<j}) + coneOut cc (sing (suc i)) ⋆⟨ C ⟩ F .F-hom + ((FinVec→Diag L α) .F-hom (singPairL {i<j = s≤s i<j}) ⋆⟨ DLCat ^op ⟩ (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) - ≡⟨ cong (λ x → coneOut cc (sing (suc i)) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ - coneOut cc (sing (suc i)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairL) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) + ≡⟨ cong (λ x → coneOut cc (sing (suc i)) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ + coneOut cc (sing (suc i)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairL) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - (coneOut cc (sing (suc i)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairL)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) + (coneOut cc (sing (suc i)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairL)) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) ≡⟨ cong - (λ x → x ⋆⟨ C ⟩ F .F-hom + (λ x → x ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) (coneOutCommutes cc singPairL) ⟩ coneOut cc (pair (suc i) (suc j) (s≤s i<j)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) ∎ + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) ∎ coneOutCommutes cc∧Suc (singPairR {i = i} {j = j} {i<j = i<j}) = coneOut cc (pair zero (suc j) (s≤s z≤)) - ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairR) - ≡⟨ cong (λ x → seq' C x (funcComp F (FinVec→Diag L β) .F-hom singPairR)) + ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairR) + ≡⟨ cong (λ x → seq' C x (funcComp F (FinVec→Diag L β) .F-hom singPairR)) (sym (coneOutCommutes cc singPairR)) ⟩ - (coneOut cc (sing (suc j)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairR)) - ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairR) + (coneOut cc (sing (suc j)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairR)) + ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairR) ≡⟨ ⋆Assoc C _ _ _ ⟩ - coneOut cc (sing (suc j)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairR) - ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairR)) - ≡⟨ cong (λ x → coneOut cc (sing (suc j)) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - coneOut cc (sing (suc j)) ⋆⟨ C ⟩ F .F-hom - ((FinVec→Diag L α) .F-hom (singPairR {i<j = s≤s z≤}) - ⋆⟨ DLCat ^op ⟩ (FinVec→Diag L β) .F-hom (singPairR {i<j = i<j})) - ≡⟨ cong (λ x → coneOut cc (sing (suc j)) ⋆⟨ C ⟩ F .F-hom x) + coneOut cc (sing (suc j)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairR) + ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L β) .F-hom singPairR)) + ≡⟨ cong (λ x → coneOut cc (sing (suc j)) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + coneOut cc (sing (suc j)) ⋆⟨ C ⟩ F .F-hom + ((FinVec→Diag L α) .F-hom (singPairR {i<j = s≤s z≤}) + ⋆⟨ DLCat ^op ⟩ (FinVec→Diag L β) .F-hom (singPairR {i<j = i<j})) + ≡⟨ cong (λ x → coneOut cc (sing (suc j)) ⋆⟨ C ⟩ F .F-hom x) (is-prop-valued _ _ _ _) ⟩ - coneOut cc (sing (suc j)) ⋆⟨ C ⟩ F .F-hom - ((FinVec→Diag L α) .F-hom (singPairR {i<j = s≤s i<j}) + coneOut cc (sing (suc j)) ⋆⟨ C ⟩ F .F-hom + ((FinVec→Diag L α) .F-hom (singPairR {i<j = s≤s i<j}) ⋆⟨ DLCat ^op ⟩ (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) - ≡⟨ cong (λ x → coneOut cc (sing (suc j)) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ - coneOut cc (sing (suc j)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairR) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) + ≡⟨ cong (λ x → coneOut cc (sing (suc j)) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ + coneOut cc (sing (suc j)) ⋆⟨ C ⟩ ((funcComp F (FinVec→Diag L α) .F-hom singPairR) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - (coneOut cc (sing (suc j)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairR)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) + (coneOut cc (sing (suc j)) ⋆⟨ C ⟩ (funcComp F (FinVec→Diag L α) .F-hom singPairR)) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) ≡⟨ cong - (λ x → x ⋆⟨ C ⟩ F .F-hom + (λ x → x ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _)))) (coneOutCommutes cc singPairR) ⟩ coneOut cc (pair (suc i) (suc j) (s≤s i<j)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) ∎ + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤RCancel _ _) (∧≤RCancel _ _))) ∎ -- our morphisms: - f : C [ c , F .F-ob (α zero) ] + f : C [ c , F .F-ob (α zero) ] f = coneOut cc (sing zero) - g : C [ c , F .F-ob (⋁ (α ∘ suc)) ] + g : C [ c , F .F-ob (⋁ (α ∘ suc)) ] g = P→L (F0=1 , presPBSq) n (α ∘ suc) c ccSuc .fst .fst k = g ⋆⟨ C ⟩ s₁ theCospan @@ -379,21 +379,21 @@ (g ⋆⟨ C ⟩ s₁ theCospan) ⋆⟨ C ⟩ (coneOut (F-cone F (⋁Cone L β)) (sing i)) ≡⟨ ⋆Assoc C _ _ _ ⟩ g ⋆⟨ C ⟩ (s₁ theCospan ⋆⟨ C ⟩ (coneOut (F-cone F (⋁Cone L β)) (sing i))) - ≡⟨ cong (λ x → g ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - g ⋆⟨ C ⟩ F .F-hom + ≡⟨ cong (λ x → g ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + g ⋆⟨ C ⟩ F .F-hom ((≤-⋁Ext _ _ λ _ → hom-∧₁ _ _) ⋆⟨ DLCat ^op ⟩ coneOut (⋁Cone L β) (sing i)) - ≡⟨ cong (λ x → g ⋆⟨ C ⟩ F .F-hom x) (is-prop-valued _ _ _ _) ⟩ - g ⋆⟨ C ⟩ F .F-hom (coneOut (⋁Cone L (α ∘ suc)) (sing i) - ⋆⟨ DLCat ^op ⟩ (FinVec→Diag L α) .F-hom (singPairR{i<j = s≤s z≤}) ) - ≡⟨ cong (λ x → g ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ + ≡⟨ cong (λ x → g ⋆⟨ C ⟩ F .F-hom x) (is-prop-valued _ _ _ _) ⟩ + g ⋆⟨ C ⟩ F .F-hom (coneOut (⋁Cone L (α ∘ suc)) (sing i) + ⋆⟨ DLCat ^op ⟩ (FinVec→Diag L α) .F-hom (singPairR{i<j = s≤s z≤}) ) + ≡⟨ cong (λ x → g ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ g ⋆⟨ C ⟩ (coneOut (F-cone F (⋁Cone L (α ∘ suc))) (sing i) - ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR) + ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ (g ⋆⟨ C ⟩ coneOut (F-cone F (⋁Cone L (α ∘ suc))) (sing i)) - ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR - ≡⟨ cong (λ x → x ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR) + ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR + ≡⟨ cong (λ x → x ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR) (P→L (F0=1 , presPBSq) n (α ∘ suc) c ccSuc .fst .snd (sing i)) ⟩ - coneOut cc (sing (suc i)) ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR + coneOut cc (sing (suc i)) ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairR ≡⟨ coneOutCommutes cc singPairR ⟩ coneOut cc (pair zero (suc i) (s≤s z≤)) ∎ @@ -405,13 +405,13 @@ singCase : ∀ i → o ⋆⟨ C ⟩ (coneOut (F-cone F (⋁Cone L β)) (sing i)) ≡ coneOut cc∧Suc (sing i) singCase i = - o ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ β i)) + o ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ β i)) ≡⟨ ⋆Assoc C _ _ _ ⟩ - f ⋆⟨ C ⟩ (s₂ theCospan ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ β i))) - ≡⟨ cong (λ x → f ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - f ⋆⟨ C ⟩ F .F-hom ((⋁IsMax _ _ λ _ → hom-∧₂ _ _) ⋆⟨ DLCat ^op ⟩ ind≤⋁ β i) - ≡⟨ cong (λ x → f ⋆⟨ C ⟩ F .F-hom x) (is-prop-valued _ _ _ _) ⟩ - f ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairL + f ⋆⟨ C ⟩ (s₂ theCospan ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ β i))) + ≡⟨ cong (λ x → f ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + f ⋆⟨ C ⟩ F .F-hom ((⋁IsMax _ _ λ _ → hom-∧₂ _ _) ⋆⟨ DLCat ^op ⟩ ind≤⋁ β i) + ≡⟨ cong (λ x → f ⋆⟨ C ⟩ F .F-hom x) (is-prop-valued _ _ _ _) ⟩ + f ⋆⟨ C ⟩ funcComp F (FinVec→Diag L α) .F-hom singPairL ≡⟨ coneOutCommutes cc singPairL ⟩ coneOut cc (pair zero (suc i) (s≤s z≤)) ∎ @@ -419,11 +419,11 @@ fgSquare = cong fst (isContr→isProp (P→L (F0=1 , presPBSq) n β c cc∧Suc) (k , isConeMorK) (o , isConeMorO)) - uniqH : ∃![ h ∈ C [ c , F .F-ob (⋁ α) ] ] + uniqH : ∃![ h ∈ C [ c , F .F-ob (⋁ α) ] ] (g ≡ h ⋆⟨ C ⟩ pbPr₁ thePB) × (f ≡ h ⋆⟨ C ⟩ pbPr₂ thePB) uniqH = univProp thePB _ _ fgSquare - toIsConeMor : ∀ (h : C [ c , F .F-ob (⋁ α) ]) + toIsConeMor : ∀ (h : C [ c , F .F-ob (⋁ α) ]) → (g ≡ h ⋆⟨ C ⟩ pbPr₁ thePB) × (f ≡ h ⋆⟨ C ⟩ pbPr₂ thePB) → isConeMor cc (F-cone F (⋁Cone L α)) h toIsConeMor h (gTriangle , fTriangle) = isConeMorSingLemma cc (F-cone F (⋁Cone L α)) singCase @@ -433,19 +433,19 @@ singCase zero = sym fTriangle singCase (suc i) = h ⋆⟨ C ⟩ (coneOut (F-cone F (⋁Cone L α)) (sing (suc i))) - ≡⟨ cong (λ x → seq' C h (F .F-hom x)) (is-prop-valued _ _ _ _) ⟩ - h ⋆⟨ C ⟩ F .F-hom (hom-∨₂ _ _ ⋆⟨ DLCat ^op ⟩ ⋁Cone L (α ∘ suc) .coneOut (sing i)) - ≡⟨ cong (seq' C h) (F .F-seq _ _) ⟩ - h ⋆⟨ C ⟩ (pbPr₁ thePB ⋆⟨ C ⟩ F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i))) + ≡⟨ cong (λ x → seq' C h (F .F-hom x)) (is-prop-valued _ _ _ _) ⟩ + h ⋆⟨ C ⟩ F .F-hom (hom-∨₂ _ _ ⋆⟨ DLCat ^op ⟩ ⋁Cone L (α ∘ suc) .coneOut (sing i)) + ≡⟨ cong (seq' C h) (F .F-seq _ _) ⟩ + h ⋆⟨ C ⟩ (pbPr₁ thePB ⋆⟨ C ⟩ F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - (h ⋆⟨ C ⟩ pbPr₁ thePB) ⋆⟨ C ⟩ F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i)) - ≡⟨ cong (λ x → x ⋆⟨ C ⟩ (F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i)))) + (h ⋆⟨ C ⟩ pbPr₁ thePB) ⋆⟨ C ⟩ F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i)) + ≡⟨ cong (λ x → x ⋆⟨ C ⟩ (F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i)))) (sym gTriangle) ⟩ - g ⋆⟨ C ⟩ F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i)) + g ⋆⟨ C ⟩ F .F-hom (⋁Cone L (α ∘ suc) .coneOut (sing i)) ≡⟨ P→L (F0=1 , presPBSq) n (α ∘ suc) c ccSuc .fst .snd (sing i) ⟩ coneOut cc (sing (suc i)) ∎ - fromIsConeMor : ∀ (h : C [ c , F .F-ob (⋁ α) ]) + fromIsConeMor : ∀ (h : C [ c , F .F-ob (⋁ α) ]) → isConeMor cc (F-cone F (⋁Cone L α)) h → (g ≡ h ⋆⟨ C ⟩ pbPr₁ thePB) × (f ≡ h ⋆⟨ C ⟩ pbPr₂ thePB) fst (fromIsConeMor h isConeMorH) = @@ -458,12 +458,12 @@ singCase : ∀ i → s ⋆⟨ C ⟩ (coneOut (F-cone F (⋁Cone L (α ∘ suc))) (sing i)) ≡ coneOut ccSuc (sing i) singCase i = - (h ⋆⟨ C ⟩ pbPr₁ thePB) ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ (α ∘ suc) i)) + (h ⋆⟨ C ⟩ pbPr₁ thePB) ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ (α ∘ suc) i)) ≡⟨ ⋆Assoc C _ _ _ ⟩ - h ⋆⟨ C ⟩ (pbPr₁ thePB ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ (α ∘ suc) i))) - ≡⟨ cong (seq' C h) (sym (F .F-seq _ _)) ⟩ - h ⋆⟨ C ⟩ F .F-hom (hom-∨₂ _ _ ⋆⟨ DLCat ^op ⟩ ind≤⋁ (α ∘ suc) i) - ≡⟨ cong (λ x → seq' C h (F .F-hom x)) (is-prop-valued _ _ _ _) ⟩ + h ⋆⟨ C ⟩ (pbPr₁ thePB ⋆⟨ C ⟩ (F .F-hom (ind≤⋁ (α ∘ suc) i))) + ≡⟨ cong (seq' C h) (sym (F .F-seq _ _)) ⟩ + h ⋆⟨ C ⟩ F .F-hom (hom-∨₂ _ _ ⋆⟨ DLCat ^op ⟩ ind≤⋁ (α ∘ suc) i) + ≡⟨ cong (λ x → seq' C h (F .F-hom x)) (is-prop-valued _ _ _ _) ⟩ h ⋆⟨ C ⟩ coneOut (F-cone F (⋁Cone L α)) (sing (suc i)) ≡⟨ isConeMorH (sing (suc i)) ⟩ coneOut cc (sing (suc i)) ∎ @@ -478,7 +478,7 @@ (L' : ℙ (fst L)) (hB : IsBasis L L') where open Category - open Functor + open Functor open DistLatticeStr ⦃...⦄ open SemilatticeStr ⦃...⦄ @@ -486,8 +486,8 @@ private DLCat = DistLatticeCategory L - BasisCat = ΣPropCat DLCat L' - DLBasisPreSheaf = Functor (BasisCat ^op) C + BasisCat = ΣPropCat DLCat L' + DLBasisPreSheaf = Functor (BasisCat ^op) C instance _ = snd L @@ -525,18 +525,18 @@ square in C but now F is only presheaf on BasisCat -} BFsq : (F : DLBasisPreSheaf) - → seq' C {x = F .F-ob x∨y} {y = F .F-ob y} {z = F .F-ob (x · y)} - (F .F-hom (hom-∨₂ L C (fst x) (fst y))) - (F .F-hom (hom-∧₂ L C (fst x) (fst y))) - ≡ seq' C {x = F .F-ob x∨y} {y = F .F-ob x} {z = F .F-ob (x · y)} - (F .F-hom (hom-∨₁ L C (fst x) (fst y))) - (F .F-hom (hom-∧₁ L C (fst x) (fst y))) - BFsq F = F-square F Bsq + → seq' C {x = F .F-ob x∨y} {y = F .F-ob y} {z = F .F-ob (x · y)} + (F .F-hom (hom-∨₂ L C (fst x) (fst y))) + (F .F-hom (hom-∧₂ L C (fst x) (fst y))) + ≡ seq' C {x = F .F-ob x∨y} {y = F .F-ob x} {z = F .F-ob (x · y)} + (F .F-hom (hom-∨₁ L C (fst x) (fst y))) + (F .F-hom (hom-∧₁ L C (fst x) (fst y))) + BFsq F = F-square F Bsq -- On a basis this is weaker than the definition below! isDLBasisSheafPullback : DLBasisPreSheaf → Type (ℓ-max (ℓ-max ℓ ℓ') ℓ'') - isDLBasisSheafPullback F = ((0∈L' : 0l ∈ L') → isTerminal C (F .F-ob (0l , 0∈L'))) + isDLBasisSheafPullback F = ((0∈L' : 0l ∈ L') → isTerminal C (F .F-ob (0l , 0∈L'))) × ((x y : ob BasisCat) (x∨y∈L' : fst x ∨l fst y ∈ L') → isPullback C _ _ _ (BFsq x y x∨y∈L' F)) where open condSquare @@ -561,14 +561,14 @@ open Order (DistLattice→Lattice L) open Cone - BDiag : Functor (DLShfDiag n ℓ) (BasisCat ^op) - F-ob BDiag (sing i) = α i - F-ob BDiag (pair i j _) = α i · α j -- α i ∧ α j + basis is closed under ∧ - F-hom BDiag idAr = is-refl _ - F-hom BDiag singPairL = ≤m→≤j _ _ (∧≤RCancel _ _) - F-hom BDiag singPairR = ≤m→≤j _ _ (∧≤LCancel _ _) - F-id BDiag = is-prop-valued _ _ _ _ - F-seq BDiag _ _ = is-prop-valued _ _ _ _ + BDiag : Functor (DLShfDiag n ℓ) (BasisCat ^op) + F-ob BDiag (sing i) = α i + F-ob BDiag (pair i j _) = α i · α j -- α i ∧ α j + basis is closed under ∧ + F-hom BDiag idAr = is-refl _ + F-hom BDiag singPairL = ≤m→≤j _ _ (∧≤RCancel _ _) + F-hom BDiag singPairR = ≤m→≤j _ _ (∧≤LCancel _ _) + F-id BDiag = is-prop-valued _ _ _ _ + F-seq BDiag _ _ = is-prop-valued _ _ _ _ module _ (⋁α∈L' : ⋁ (λ i → α i .fst) ∈ L') where private @@ -598,28 +598,28 @@ DLBasisSheaf→Terminal : ∀ (F : DLBasisPreSheaf) → isDLBasisSheaf F - → ∀ (0∈L' : 0l ∈ L') → isTerminal C (F .F-ob (0l , 0∈L')) + → ∀ (0∈L' : 0l ∈ L') → isTerminal C (F .F-ob (0l , 0∈L')) DLBasisSheaf→Terminal F isSheafF 0∈L' = isTerminalF0 where open Cone open condCone {n = 0} (λ ()) emptyCone = B⋁Cone 0∈L' - isLimConeF0 : isLimCone _ (F .F-ob (0l , 0∈L')) (F-cone F emptyCone) + isLimConeF0 : isLimCone _ (F .F-ob (0l , 0∈L')) (F-cone F emptyCone) isLimConeF0 = isSheafF (λ ()) 0∈L' - toCone : (y : ob C) → Cone (funcComp F BDiag) y + toCone : (y : ob C) → Cone (funcComp F BDiag) y coneOut (toCone y) (sing ()) coneOut (toCone y) (pair () _ _) coneOutCommutes (toCone y) {u = sing ()} idAr coneOutCommutes (toCone y) {u = pair () _ _} idAr - toConeMor : ∀ (y : ob C) (f : C [ y , F .F-ob (0l , 0∈L') ]) + toConeMor : ∀ (y : ob C) (f : C [ y , F .F-ob (0l , 0∈L') ]) → isConeMor (toCone y) (F-cone F emptyCone) f toConeMor y f (sing ()) toConeMor y f (pair () _ _) - isTerminalF0 : isTerminal C (F .F-ob (0l , 0∈L')) + isTerminalF0 : isTerminal C (F .F-ob (0l , 0∈L')) fst (isTerminalF0 y) = isLimConeF0 _ (toCone y) .fst .fst snd (isTerminalF0 y) f = cong fst (isLimConeF0 _ (toCone y) .snd (_ , toConeMor y f)) \ No newline at end of file diff --git a/Cubical.Categories.DistLatticeSheaf.ComparisonLemma.html b/Cubical.Categories.DistLatticeSheaf.ComparisonLemma.html index 446c0a6e3e..b7144b7626 100644 --- a/Cubical.Categories.DistLatticeSheaf.ComparisonLemma.html +++ b/Cubical.Categories.DistLatticeSheaf.ComparisonLemma.html @@ -66,7 +66,7 @@ open Category hiding (_∘_) - open Functor + open Functor open NatTrans open Cone open LimCone @@ -85,51 +85,51 @@ _ = snd L Lᵒᵖ = DistLatticeCategory L ^op - Bᵒᵖ = ΣPropCat (DistLatticeCategory L) B ^op + Bᵒᵖ = ΣPropCat (DistLatticeCategory L) B ^op - SheafB = ΣPropCat (FUNCTOR Bᵒᵖ C) isDLBasisSheafProp - SheafL = ΣPropCat (FUNCTOR Lᵒᵖ C) (isDLSheafProp L C) + SheafB = ΣPropCat (FUNCTOR Bᵒᵖ C) isDLBasisSheafProp + SheafL = ΣPropCat (FUNCTOR Lᵒᵖ C) (isDLSheafProp L C) - i = baseIncl ^opF + i = baseIncl ^opF - restPresSheafProp : ∀ (F : Functor Lᵒᵖ C) → isDLSheaf L C F → isDLBasisSheaf (F ∘F i) + restPresSheafProp : ∀ (F : Functor Lᵒᵖ C) → isDLSheaf L C F → isDLBasisSheaf (F ∘F i) restPresSheafProp F isSheafF α ⋁α∈B = - transport (λ i → isLimCone (diagPath i) (F .F-ob (⋁ α')) (limConesPathP i)) (isSheafF _ α') + transport (λ i → isLimCone (diagPath i) (F .F-ob (⋁ α')) (limConesPathP i)) (isSheafF _ α') where open condCone α α' : FinVec (fst L) _ α' i = α i .fst - diagPath : F ∘F (FinVec→Diag L α') ≡ (F ∘F i) ∘F BDiag - diagPath = Functor≡ diagPathOb diagPathHom + diagPath : F ∘F (FinVec→Diag L α') ≡ (F ∘F i) ∘F BDiag + diagPath = Functor≡ diagPathOb diagPathHom where - diagPathOb : ∀ c → (F ∘F (FinVec→Diag L α')) .F-ob c ≡ ((F ∘F i) ∘F BDiag) .F-ob c + diagPathOb : ∀ c → (F ∘F (FinVec→Diag L α')) .F-ob c ≡ ((F ∘F i) ∘F BDiag) .F-ob c diagPathOb (sing i) = refl - diagPathOb (pair i j i<j) = cong (F .F-ob) (∧lComm _ _) + diagPathOb (pair i j i<j) = cong (F .F-ob) (∧lComm _ _) diagPathHom : ∀ {c} {d} f → PathP (λ i → C [ diagPathOb c i , diagPathOb d i ]) - ((F ∘F (FinVec→Diag L α')) .F-hom f) - (((F ∘F i) ∘F BDiag) .F-hom f) + ((F ∘F (FinVec→Diag L α')) .F-hom f) + (((F ∘F i) ∘F BDiag) .F-hom f) diagPathHom {sing i} idAr = refl diagPathHom {pair i j i<j} idAr = functorCongP {F = F} (toPathP (is-prop-valued _ _ _ _)) diagPathHom singPairL = functorCongP {F = F} (toPathP (is-prop-valued _ _ _ _)) diagPathHom singPairR = functorCongP {F = F} (toPathP (is-prop-valued _ _ _ _)) - limConesPathP : PathP (λ i → Cone (diagPath i) (F .F-ob (⋁ α'))) - (F-cone F (⋁Cone L α')) (F-cone (F ∘F i) (B⋁Cone ⋁α∈B)) + limConesPathP : PathP (λ i → Cone (diagPath i) (F .F-ob (⋁ α'))) + (F-cone F (⋁Cone L α')) (F-cone (F ∘F i) (B⋁Cone ⋁α∈B)) limConesPathP = conePathP limConesPathPOb where - limConesPathPOb : ∀ c → PathP (λ i → C [ F .F-ob (⋁ α') , diagPath i .F-ob c ]) + limConesPathPOb : ∀ c → PathP (λ i → C [ F .F-ob (⋁ α') , diagPath i .F-ob c ]) (F-cone F (⋁Cone L α') .coneOut c) - (F-cone (F ∘F i) (B⋁Cone ⋁α∈B) .coneOut c) + (F-cone (F ∘F i) (B⋁Cone ⋁α∈B) .coneOut c) limConesPathPOb (sing i) = refl limConesPathPOb (pair i j i<j) = functorCongP {F = F} (toPathP (is-prop-valued _ _ _ _)) --restriction to basis as functor - sheafRestriction : Functor SheafL SheafB - sheafRestriction = ΣPropCatFunc (precomposeF C i) restPresSheafProp + sheafRestriction : Functor SheafL SheafB + sheafRestriction = ΣPropCatFunc (precomposeF C i) restPresSheafProp -- important lemma: a natural transformation between sheaves on L is an -- iso if the restriction to B is an iso. This will give us that @@ -141,8 +141,8 @@ restIsoLemma (F , isSheafF) (G , isSheafG) α αiIso x = PT.rec (isPropIsIso _) basisHyp (isBasisB .⋁Basis x) where - Fi = F ∘F i - Gi = G ∘F i + Fi = F ∘F i + Gi = G ∘F i open NatIso αiNatIso : NatIso Fi Gi trans αiNatIso = α ∘ˡ i @@ -163,64 +163,64 @@ uᴮDiag = condCone.BDiag uᴮ - αi⁻¹ : (v : ob Bᵒᵖ) → C [ Gi .F-ob v , Fi .F-ob v ] + αi⁻¹ : (v : ob Bᵒᵖ) → C [ Gi .F-ob v , Fi .F-ob v ] αi⁻¹ v = αiIso v .inv - σ : NatTrans (F ∘F (FinVec→Diag L u)) (G ∘F (FinVec→Diag L u)) - N-ob σ = α .N-ob ∘ FinVec→Diag L u .F-ob - N-hom σ = α .N-hom ∘ FinVec→Diag L u .F-hom + σ : NatTrans (F ∘F (FinVec→Diag L u)) (G ∘F (FinVec→Diag L u)) + N-ob σ = α .N-ob ∘ FinVec→Diag L u .F-ob + N-hom σ = α .N-hom ∘ FinVec→Diag L u .F-hom open SemilatticeStr ⦃...⦄ instance _ = snd (Basis→MeetSemilattice L B isBasisB) - σ⁻¹ : NatTrans (G ∘F (FinVec→Diag L u)) (F ∘F (FinVec→Diag L u)) - N-ob σ⁻¹ (sing i) = αi⁻¹ (uᴮDiag .F-ob (sing i)) + σ⁻¹ : NatTrans (G ∘F (FinVec→Diag L u)) (F ∘F (FinVec→Diag L u)) + N-ob σ⁻¹ (sing i) = αi⁻¹ (uᴮDiag .F-ob (sing i)) N-ob σ⁻¹ (pair i j i<j) = αi⁻¹ ((u j , u∈B j) · (u i , u∈B i)) -- (uᴮDiag .F-ob (pair i j i<j)) modulo swapping i and j N-hom σ⁻¹ (idAr {x = v}) = - G .F-hom (id Lᵒᵖ) ⋆⟨ C ⟩ σ⁻¹ .N-ob v ≡⟨ cong (λ f → f ⋆⟨ C ⟩ σ⁻¹ .N-ob v) (G .F-id) ⟩ + G .F-hom (id Lᵒᵖ) ⋆⟨ C ⟩ σ⁻¹ .N-ob v ≡⟨ cong (λ f → f ⋆⟨ C ⟩ σ⁻¹ .N-ob v) (G .F-id) ⟩ id C ⋆⟨ C ⟩ σ⁻¹ .N-ob v ≡⟨ ⋆IdL C _ ⟩ σ⁻¹ .N-ob v ≡⟨ sym (⋆IdR C _) ⟩ - σ⁻¹ .N-ob v ⋆⟨ C ⟩ id C ≡⟨ cong (λ f → σ⁻¹ .N-ob v ⋆⟨ C ⟩ f) (sym (F .F-id)) ⟩ - σ⁻¹ .N-ob v ⋆⟨ C ⟩ F .F-hom (id Lᵒᵖ) ∎ + σ⁻¹ .N-ob v ⋆⟨ C ⟩ id C ≡⟨ cong (λ f → σ⁻¹ .N-ob v ⋆⟨ C ⟩ f) (sym (F .F-id)) ⟩ + σ⁻¹ .N-ob v ⋆⟨ C ⟩ F .F-hom (id Lᵒᵖ) ∎ N-hom σ⁻¹ (singPairL {i} {j} {i<j}) = transport (λ 𝕚 → p 𝕚 ≡ r 𝕚) q where - p : PathP (λ 𝕚 → C [ G .F-ob (u i) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) - (G .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j))) - (G .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ ((u j , u∈B j) · (u i , u∈B i))) - p 𝕚 = G .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u i)) + p : PathP (λ 𝕚 → C [ G .F-ob (u i) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) + (G .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j))) + (G .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ ((u j , u∈B j) · (u i , u∈B i))) + p 𝕚 = G .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u i)) (≤m→≤j _ _ (∧≤RCancel _ _)) (≤m→≤j _ _ (∧≤LCancel _ _)) 𝕚) ⋆⟨ C ⟩ αi⁻¹ (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚) - q : G .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j)) - ≡ αi⁻¹ (u i , u∈B i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + q : G .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j)) + ≡ αi⁻¹ (u i , u∈B i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) q = sqLL αiNatIso - r : PathP (λ 𝕚 → C [ G .F-ob (u i) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) - (αi⁻¹ (u i , u∈B i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _))) - (αi⁻¹ (u i , u∈B i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) + r : PathP (λ 𝕚 → C [ G .F-ob (u i) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) + (αi⁻¹ (u i , u∈B i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _))) + (αi⁻¹ (u i , u∈B i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) r 𝕚 = αi⁻¹ (u i , u∈B i) - ⋆⟨ C ⟩ F .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u i)) + ⋆⟨ C ⟩ F .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u i)) (≤m→≤j _ _ (∧≤RCancel _ _)) (≤m→≤j _ _ (∧≤LCancel _ _)) 𝕚) N-hom σ⁻¹ (singPairR {i} {j} {i<j}) = transport (λ 𝕚 → p 𝕚 ≡ r 𝕚) q where - p : PathP (λ 𝕚 → C [ G .F-ob (u j) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) - (G .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j))) - (G .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ ((u j , u∈B j) · (u i , u∈B i))) - p 𝕚 = G .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u j)) + p : PathP (λ 𝕚 → C [ G .F-ob (u j) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) + (G .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j))) + (G .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ ((u j , u∈B j) · (u i , u∈B i))) + p 𝕚 = G .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u j)) (≤m→≤j _ _ (∧≤LCancel _ _)) (≤m→≤j _ _ (∧≤RCancel _ _)) 𝕚) ⋆⟨ C ⟩ αi⁻¹ (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚) - q : G .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j)) - ≡ αi⁻¹ (u j , u∈B j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + q : G .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ⋆⟨ C ⟩ αi⁻¹ (uᴮDiag .F-ob (pair i j i<j)) + ≡ αi⁻¹ (u j , u∈B j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) q = sqLL αiNatIso - r : PathP (λ 𝕚 → C [ G .F-ob (u j) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) - (αi⁻¹ (u j , u∈B j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) - (αi⁻¹ (u j , u∈B j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _))) + r : PathP (λ 𝕚 → C [ G .F-ob (u j) , F .F-ob (fst (·Comm (u i , u∈B i) (u j , u∈B j) 𝕚)) ]) + (αi⁻¹ (u j , u∈B j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) + (αi⁻¹ (u j , u∈B j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _))) r 𝕚 = αi⁻¹ (u j , u∈B j) - ⋆⟨ C ⟩ F .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u j)) + ⋆⟨ C ⟩ F .F-hom (isProp→PathP (λ 𝕚' → is-prop-valued (∧lComm (u i) (u j) 𝕚') (u j)) (≤m→≤j _ _ (∧≤LCancel _ _)) (≤m→≤j _ _ (∧≤RCancel _ _)) 𝕚) -- σ and σ⁻¹ are inverse: @@ -257,12 +257,12 @@ (isConeMorSingLemma (limOfArrowsCone FLimCone σ) (F-cone G (⋁Cone L u)) λ i → sym (α .N-hom (ind≤⋁ u i))) - q : PathP (λ i → C [ F .F-ob (⋁u≡x i) , G .F-ob (⋁u≡x i) ]) (α .N-ob (⋁ u)) (α .N-ob x) + q : PathP (λ i → C [ F .F-ob (⋁u≡x i) , G .F-ob (⋁u≡x i) ]) (α .N-ob (⋁ u)) (α .N-ob x) q = cong (α .N-ob) ⋁u≡x -- notation - private module _ {F G : Functor Bᵒᵖ C} (α : NatTrans F G) (x : ob Lᵒᵖ) where + private module _ {F G : Functor Bᵒᵖ C} (α : NatTrans F G) (x : ob Lᵒᵖ) where theDiag = (_↓Diag limitC i F x) -- note that (_↓Diag limitC i F x) = (_↓Diag limitC i G x) definitionally FLimCone = limitC (_↓Diag limitC i F x) (T* limitC i F x) @@ -280,13 +280,13 @@ coneOut diagCone (u , y≥u) = limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ α .N-ob u coneOutCommutes diagCone {u = (u , y≥u)} {v = (v , y≥v)} (u≥v , _) = - (limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ α .N-ob u) ⋆⟨ C ⟩ G .F-hom u≥v + (limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ α .N-ob u) ⋆⟨ C ⟩ G .F-hom u≥v ≡⟨ ⋆Assoc C _ _ _ ⟩ - limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ (α .N-ob u ⋆⟨ C ⟩ G .F-hom u≥v) + limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ (α .N-ob u ⋆⟨ C ⟩ G .F-hom u≥v) ≡⟨ cong (seq' C (limOut FLimCone (u , is-trans _ _ _ y≥u x≥y))) (sym (α .N-hom u≥v)) ⟩ - limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ (F .F-hom u≥v ⋆⟨ C ⟩ α .N-ob v) + limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ (F .F-hom u≥v ⋆⟨ C ⟩ α .N-ob v) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - (limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ F .F-hom u≥v) ⋆⟨ C ⟩ α .N-ob v + (limOut FLimCone (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ F .F-hom u≥v) ⋆⟨ C ⟩ α .N-ob v ≡⟨ cong (λ x → x ⋆⟨ C ⟩ α .N-ob v) (limOutCommutes FLimCone (u≥v , is-prop-valued _ _ _ _)) ⟩ limOut FLimCone (v , is-trans _ _ _ y≥v x≥y) ⋆⟨ C ⟩ α .N-ob v ∎ diagArrow : C [ RanOb limitC i F x , RanOb limitC i G y ] @@ -294,10 +294,10 @@ - DLRanFun : Functor (FUNCTOR Bᵒᵖ C) (FUNCTOR Lᵒᵖ C) - F-ob DLRanFun = DLRan - N-ob (F-hom DLRanFun α) x = limOfArrows (FLimCone α _) (GLimCone α _) (↓nt α x) - N-hom (F-hom DLRanFun {x = F} {y = G} α) {x = x} {y = y} x≥y = + DLRanFun : Functor (FUNCTOR Bᵒᵖ C) (FUNCTOR Lᵒᵖ C) + F-ob DLRanFun = DLRan + N-ob (F-hom DLRanFun α) x = limOfArrows (FLimCone α _) (GLimCone α _) (↓nt α x) + N-hom (F-hom DLRanFun {x = F} {y = G} α) {x = x} {y = y} x≥y = sym (limArrowUnique (GLimCone α y) _ (diagCone α x y x≥y) _ isConeMorL) ∙ (limArrowUnique (GLimCone α y) _ _ _ isConeMorR) where @@ -339,17 +339,17 @@ limOut (FLimCone α x) (u , is-trans _ _ _ y≥u x≥y) ⋆⟨ C ⟩ α .N-ob u ∎ - F-id DLRanFun {x = F} = makeNatTransPath + F-id DLRanFun {x = F} = makeNatTransPath (funExt λ x → limOfArrowsId (FLimCone (idTrans F) x)) - F-seq DLRanFun α β = makeNatTransPath + F-seq DLRanFun α β = makeNatTransPath (funExt λ x → limOfArrowsSeq (FLimCone α x) (GLimCone α x) (GLimCone β x) (↓nt α x) (↓nt β x)) --extension of sheaves as functor - sheafExtension : Functor SheafB SheafL - sheafExtension = ΣPropCatFunc DLRanFun (isDLSheafDLRan isBasisB) + sheafExtension : Functor SheafB SheafL + sheafExtension = ΣPropCatFunc DLRanFun (isDLSheafDLRan isBasisB) @@ -365,7 +365,7 @@ -- the unit is induced by the universal property N-ob (trans (η winv)) (F , _ ) = - DLRanUnivProp (F ∘F i) F (idTrans _) .fst .fst + DLRanUnivProp (F ∘F i) F (idTrans _) .fst .fst N-hom (trans (η winv)) {x = (F , _)} {y = (G , _)} α = makeNatTransPath (funExt goal) where @@ -385,13 +385,13 @@ (NatTransCone _ _ _ G (idTrans _) x) ⋆⟨ C ⟩ limOut (GLimCone (α ∘ˡ i) _) u⇂) ≡⟨ cong (λ y → α .N-ob x ⋆⟨ C ⟩ y) (limArrowCommutes (GLimCone (α ∘ˡ i) _) _ _ _) ⟩ - α .N-ob x ⋆⟨ C ⟩ (G .F-hom u≤x ⋆⟨ C ⟩ id C) + α .N-ob x ⋆⟨ C ⟩ (G .F-hom u≤x ⋆⟨ C ⟩ id C) ≡⟨ cong (λ y → α .N-ob x ⋆⟨ C ⟩ y) (⋆IdR C _) ⟩ - α .N-ob x ⋆⟨ C ⟩ G .F-hom u≤x + α .N-ob x ⋆⟨ C ⟩ G .F-hom u≤x ≡⟨ sym (α .N-hom u≤x) ⟩ - F .F-hom u≤x ⋆⟨ C ⟩ α .N-ob u + F .F-hom u≤x ⋆⟨ C ⟩ α .N-ob u ≡⟨ cong (λ x → x ⋆⟨ C ⟩ α .N-ob u) (sym (⋆IdR C _)) ⟩ - F .F-hom u≤x ⋆⟨ C ⟩ id C ⋆⟨ C ⟩ α .N-ob u ∎ + F .F-hom u≤x ⋆⟨ C ⟩ id C ⋆⟨ C ⟩ α .N-ob u ∎ goal : ∀ (x : ob Lᵒᵖ) → α .N-ob x ⋆⟨ C ⟩ limArrow (GLimCone (α ∘ˡ i) _) _ @@ -403,33 +403,33 @@ goal x = sym (limArrowUnique _ _ _ _ (isConeMorComp x)) ∙ limArrowCompLimOfArrows _ _ _ _ _ - nIso (η winv) (F , isSheafF) = isIsoΣPropCat _ _ _ + nIso (η winv) (F , isSheafF) = isIsoΣPropCat _ _ _ (NatIso→FUNCTORIso _ _ σNatIso .snd) where - σ = DLRanUnivProp (F ∘F i) F (idTrans _) .fst .fst + σ = DLRanUnivProp (F ∘F i) F (idTrans _) .fst .fst σRestIso : isIso (FUNCTOR Bᵒᵖ C) (σ ∘ˡ i) - inv σRestIso = DLRanNatTrans (F ∘F i) - sec σRestIso = let ε = DLRanNatTrans (F ∘F i) - ε⁻¹ = NatIso→FUNCTORIso _ _ (DLRanNatIso (F ∘F i)) .snd .inv + inv σRestIso = DLRanNatTrans (F ∘F i) + sec σRestIso = let ε = DLRanNatTrans (F ∘F i) + ε⁻¹ = NatIso→FUNCTORIso _ _ (DLRanNatIso (F ∘F i)) .snd .inv in ε ●ᵛ (σ ∘ˡ i) ≡⟨ cong (λ x → ε ●ᵛ x) (sym (⋆IdR (FUNCTOR Bᵒᵖ C) _)) ⟩ ε ●ᵛ ((σ ∘ˡ i) ●ᵛ idTrans _) ≡⟨ cong (λ x → ε ●ᵛ ((σ ∘ˡ i) ●ᵛ x)) - (sym (NatIso→FUNCTORIso _ _ (DLRanNatIso (F ∘F i)) .snd .ret)) ⟩ + (sym (NatIso→FUNCTORIso _ _ (DLRanNatIso (F ∘F i)) .snd .ret)) ⟩ ε ●ᵛ ((σ ∘ˡ i) ●ᵛ (ε ●ᵛ ε⁻¹)) ≡⟨ cong (λ x → ε ●ᵛ x) (sym (⋆Assoc (FUNCTOR Bᵒᵖ C) _ _ _)) ⟩ ε ●ᵛ ((σ ∘ˡ i) ●ᵛ ε ●ᵛ ε⁻¹) ≡⟨ cong (λ x → ε ●ᵛ (x ●ᵛ ε⁻¹)) - (sym (DLRanUnivProp (F ∘F i) F (idTrans _) .fst .snd)) ⟩ + (sym (DLRanUnivProp (F ∘F i) F (idTrans _) .fst .snd)) ⟩ ε ●ᵛ (idTrans _ ●ᵛ ε⁻¹) ≡⟨ cong (λ x → ε ●ᵛ x) (⋆IdL (FUNCTOR Bᵒᵖ C) _) ⟩ ε ●ᵛ ε⁻¹ - ≡⟨ NatIso→FUNCTORIso _ _ (DLRanNatIso (F ∘F i)) .snd .ret ⟩ + ≡⟨ NatIso→FUNCTORIso _ _ (DLRanNatIso (F ∘F i)) .snd .ret ⟩ idTrans _ ∎ - ret σRestIso = sym (DLRanUnivProp (F ∘F i) F (idTrans _) .fst .snd) + ret σRestIso = sym (DLRanUnivProp (F ∘F i) F (idTrans _) .fst .snd) - σNatIso : NatIso F (DLRan (F ∘F i)) + σNatIso : NatIso F (DLRan (F ∘F i)) trans σNatIso = σ nIso σNatIso = restIsoLemma (F , isSheafF) @@ -442,7 +442,7 @@ N-hom (trans (ε winv)) α = -- DLRanNatTrans F is functorial in F makeNatTransPath (funExt (λ u → limOfArrowsOut (FLimCone α (u .fst)) (GLimCone α (u .fst)) _ _)) - nIso (ε winv) (F , isSheafF) = isIsoΣPropCat _ _ _ + nIso (ε winv) (F , isSheafF) = isIsoΣPropCat _ _ _ (NatIso→FUNCTORIso _ _ (DLRanNatIso F) .snd) diff --git a/Cubical.Categories.DistLatticeSheaf.Diagram.html b/Cubical.Categories.DistLatticeSheaf.Diagram.html index f15e547385..760007e542 100644 --- a/Cubical.Categories.DistLatticeSheaf.Diagram.html +++ b/Cubical.Categories.DistLatticeSheaf.Diagram.html @@ -128,9 +128,9 @@ -- a lemma for eliminating pair cases -- when checking that somthing is a cone morphism -module _ {C : Category ℓ ℓ'} {n : ℕ} {F : Functor (DLShfDiag n ℓ'') C} where +module _ {C : Category ℓ ℓ'} {n : ℕ} {F : Functor (DLShfDiag n ℓ'') C} where open Category - open Functor F + open Functor F open Cone isConeMorSingLemma : {c d : ob C} {f : C [ c , d ]} @@ -141,11 +141,11 @@ isConeMorSingLemma {f = f} cc cd singHyp (pair i j i<j) = f ⋆⟨ C ⟩ coneOut cd (pair i j i<j) ≡⟨ cong (λ x → f ⋆⟨ C ⟩ x) (sym (cd .coneOutCommutes singPairL)) ⟩ - f ⋆⟨ C ⟩ (coneOut cd (sing i) ⋆⟨ C ⟩ F-hom singPairL) + f ⋆⟨ C ⟩ (coneOut cd (sing i) ⋆⟨ C ⟩ F-hom singPairL) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - (f ⋆⟨ C ⟩ coneOut cd (sing i)) ⋆⟨ C ⟩ F-hom singPairL - ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F-hom singPairL) (singHyp i) ⟩ - coneOut cc (sing i) ⋆⟨ C ⟩ F-hom singPairL + (f ⋆⟨ C ⟩ coneOut cd (sing i)) ⋆⟨ C ⟩ F-hom singPairL + ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F-hom singPairL) (singHyp i) ⟩ + coneOut cc (sing i) ⋆⟨ C ⟩ F-hom singPairL ≡⟨ cc .coneOutCommutes singPairL ⟩ coneOut cc (pair i j i<j) ∎ @@ -166,18 +166,18 @@ open Order (DistLattice→Lattice L') open Category LCat - open Functor + open Functor open Cone - FinVec→Diag : {n : ℕ} → FinVec L n → Functor (DLShfDiag n ℓ) LCat - F-ob (FinVec→Diag α) (sing i) = α i - F-ob (FinVec→Diag α) (pair i j _) = α j ∧l α i - F-hom (FinVec→Diag α) idAr = is-refl _ - F-hom (FinVec→Diag α) singPairL = ≤m→≤j _ _ (∧≤LCancel _ _) - F-hom (FinVec→Diag α) singPairR = ≤m→≤j _ _ (∧≤RCancel _ _) - F-id (FinVec→Diag α) = is-prop-valued _ _ _ _ - F-seq (FinVec→Diag α) _ _ = is-prop-valued _ _ _ _ + FinVec→Diag : {n : ℕ} → FinVec L n → Functor (DLShfDiag n ℓ) LCat + F-ob (FinVec→Diag α) (sing i) = α i + F-ob (FinVec→Diag α) (pair i j _) = α j ∧l α i + F-hom (FinVec→Diag α) idAr = is-refl _ + F-hom (FinVec→Diag α) singPairL = ≤m→≤j _ _ (∧≤LCancel _ _) + F-hom (FinVec→Diag α) singPairR = ≤m→≤j _ _ (∧≤RCancel _ _) + F-id (FinVec→Diag α) = is-prop-valued _ _ _ _ + F-seq (FinVec→Diag α) _ _ = is-prop-valued _ _ _ _ ⋁Cone : {n : ℕ} (α : FinVec L n) → Cone (FinVec→Diag α) (⋁ α) coneOut (⋁Cone α) (sing i) = ind≤⋁ α i @@ -187,7 +187,7 @@ isLimCone⋁Cone : {n : ℕ} (α : FinVec L n) → isLimCone (FinVec→Diag α) (⋁ α) (⋁Cone α) fst (fst (isLimCone⋁Cone α u uCone)) = ⋁IsMax α _ λ i → uCone .coneOut (sing i) snd (fst (isLimCone⋁Cone α u uCone)) _ = is-prop-valued _ _ _ _ - snd (isLimCone⋁Cone α _ uCone) _ = Σ≡Prop (λ _ → isPropIsConeMor uCone (⋁Cone α) _) + snd (isLimCone⋁Cone α _ uCone) _ = Σ≡Prop (λ _ → isPropIsConeMor uCone (⋁Cone α) _) (is-prop-valued _ _ _ _) @@ -196,7 +196,7 @@ (cspan : Cospan C) (pback : Pullback C cspan) where - open Functor + open Functor open Cone open Cospan ⦃...⦄ open Pullback ⦃...⦄ @@ -204,21 +204,21 @@ _ = cspan _ = pback - cospanAsDiag : Functor (DLShfDiag 2 ℓ'') C - F-ob cospanAsDiag (sing zero) = l - F-ob cospanAsDiag (sing one) = r - F-ob cospanAsDiag (pair _ _ _) = m - F-hom cospanAsDiag idAr = id C - F-hom cospanAsDiag {x = sing zero} singPairL = s₁ - F-hom cospanAsDiag {x = sing one} singPairL = s₂ - F-hom cospanAsDiag {x = sing zero} singPairR = s₁ - F-hom cospanAsDiag {x = sing one} singPairR = s₂ - F-id cospanAsDiag = refl - F-seq cospanAsDiag idAr idAr = sym (⋆IdL C _) - F-seq cospanAsDiag idAr singPairL = sym (⋆IdL C _) - F-seq cospanAsDiag idAr singPairR = sym (⋆IdL C _) - F-seq cospanAsDiag singPairL idAr = sym (⋆IdR C _) - F-seq cospanAsDiag singPairR idAr = sym (⋆IdR C _) + cospanAsDiag : Functor (DLShfDiag 2 ℓ'') C + F-ob cospanAsDiag (sing zero) = l + F-ob cospanAsDiag (sing one) = r + F-ob cospanAsDiag (pair _ _ _) = m + F-hom cospanAsDiag idAr = id C + F-hom cospanAsDiag {x = sing zero} singPairL = s₁ + F-hom cospanAsDiag {x = sing one} singPairL = s₂ + F-hom cospanAsDiag {x = sing zero} singPairR = s₁ + F-hom cospanAsDiag {x = sing one} singPairR = s₂ + F-id cospanAsDiag = refl + F-seq cospanAsDiag idAr idAr = sym (⋆IdL C _) + F-seq cospanAsDiag idAr singPairL = sym (⋆IdL C _) + F-seq cospanAsDiag idAr singPairR = sym (⋆IdL C _) + F-seq cospanAsDiag singPairL idAr = sym (⋆IdR C _) + F-seq cospanAsDiag singPairR idAr = sym (⋆IdR C _) pbPrAsCone : Cone cospanAsDiag pbOb coneOut pbPrAsCone (sing zero) = pbPr₁ @@ -282,14 +282,14 @@ module DLShfDiagsAsPullbacks (C : Category ℓ ℓ') - (F : Functor (DLShfDiag 2 ℓ'') C) + (F : Functor (DLShfDiag 2 ℓ'') C) (limF : LimCone F) where open Cospan open Pullback - open Functor ⦃...⦄ + open Functor ⦃...⦄ open Cone ⦃...⦄ open LimCone ⦃...⦄ instance @@ -299,11 +299,11 @@ DiagAsCospan : Cospan C - l DiagAsCospan = F-ob (sing zero) - m DiagAsCospan = F-ob (pair zero one (s≤s z≤)) - r DiagAsCospan = F-ob (sing one) - s₁ DiagAsCospan = F-hom singPairL - s₂ DiagAsCospan = F-hom singPairR + l DiagAsCospan = F-ob (sing zero) + m DiagAsCospan = F-ob (pair zero one (s≤s z≤)) + r DiagAsCospan = F-ob (sing one) + s₁ DiagAsCospan = F-hom singPairL + s₂ DiagAsCospan = F-hom singPairR LimAsPullback : Pullback C DiagAsCospan pbOb LimAsPullback = lim @@ -324,7 +324,7 @@ Cone.coneOut theCone (pair zero one (s≤s z≤)) = f ⋆⟨ C ⟩ DiagAsCospan .s₁ Cone.coneOut theCone (pair one zero ()) Cone.coneOut theCone (pair one one (s≤s ())) - Cone.coneOutCommutes theCone {u} idAr = cong (seq' C (Cone.coneOut theCone u)) F-id + Cone.coneOutCommutes theCone {u} idAr = cong (seq' C (Cone.coneOut theCone u)) F-id ∙ ⋆IdR C (Cone.coneOut theCone u) Cone.coneOutCommutes theCone {sing zero} {pair ._ one (s≤s z≤)} singPairL = refl Cone.coneOutCommutes theCone {sing one} {pair ._ one (s≤s ())} singPairL @@ -342,13 +342,13 @@ toConeMor h' (tr₁ , tr₂) (pair zero one (s≤s z≤)) = path where path : h' ⋆⟨ C ⟩ coneOut (pair zero one (s≤s z≤)) - ≡ f ⋆⟨ C ⟩ F-hom singPairL + ≡ f ⋆⟨ C ⟩ F-hom singPairL path = h' ⋆⟨ C ⟩ coneOut (pair zero one (s≤s z≤)) ≡⟨ cong (seq' C h') (sym (coneOutCommutes singPairL)) ⟩ - h' ⋆⟨ C ⟩ (coneOut (sing zero) ⋆⟨ C ⟩ F-hom singPairL) + h' ⋆⟨ C ⟩ (coneOut (sing zero) ⋆⟨ C ⟩ F-hom singPairL) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - (h' ⋆⟨ C ⟩ coneOut (sing zero)) ⋆⟨ C ⟩ F-hom singPairL - ≡⟨ cong (λ x → seq' C x (F-hom singPairL)) (sym tr₁) ⟩ - f ⋆⟨ C ⟩ F-hom singPairL ∎ + (h' ⋆⟨ C ⟩ coneOut (sing zero)) ⋆⟨ C ⟩ F-hom singPairL + ≡⟨ cong (λ x → seq' C x (F-hom singPairL)) (sym tr₁) ⟩ + f ⋆⟨ C ⟩ F-hom singPairL ∎ toConeMor h' (tr₁ , tr₂) (pair one one (s≤s ())) \ No newline at end of file diff --git a/Cubical.Categories.DistLatticeSheaf.Extension.html b/Cubical.Categories.DistLatticeSheaf.Extension.html index b28ba7dc35..1395ae9af8 100644 --- a/Cubical.Categories.DistLatticeSheaf.Extension.html +++ b/Cubical.Categories.DistLatticeSheaf.Extension.html @@ -57,33 +57,33 @@ (limitC : Limits {ℓ} {ℓ} C) (L' : ℙ (fst L)) where open Category hiding (_∘_) - open Functor + open Functor open Cone open LimCone private DLCat = DistLatticeCategory L - DLSubCat = ΣPropCat DLCat L' - DLPreSheaf = Functor (DLCat ^op) C - DLSubPreSheaf = Functor (DLSubCat ^op) C + DLSubCat = ΣPropCat DLCat L' + DLPreSheaf = Functor (DLCat ^op) C + DLSubPreSheaf = Functor (DLSubCat ^op) C - baseIncl : Functor DLSubCat DLCat - F-ob baseIncl = fst - F-hom baseIncl f = f - F-id baseIncl = refl - F-seq baseIncl _ _ = refl + baseIncl : Functor DLSubCat DLCat + F-ob baseIncl = fst + F-hom baseIncl f = f + F-id baseIncl = refl + F-seq baseIncl _ _ = refl DLRan : DLSubPreSheaf → DLPreSheaf - DLRan = Ran limitC (baseIncl ^opF) + DLRan = Ran limitC (baseIncl ^opF) - DLRanNatTrans : (F : DLSubPreSheaf) → NatTrans (funcComp (DLRan F) (baseIncl ^opF)) F + DLRanNatTrans : (F : DLSubPreSheaf) → NatTrans (funcComp (DLRan F) (baseIncl ^opF)) F DLRanNatTrans = RanNatTrans _ _ - DLRanUnivProp : ∀ (F : DLSubPreSheaf) (G : DLPreSheaf) (α : NatTrans (G ∘F (baseIncl ^opF)) F) - → ∃![ σ ∈ NatTrans G (DLRan F) ] α ≡ (σ ∘ˡ (baseIncl ^opF)) ●ᵛ (DLRanNatTrans F) + DLRanUnivProp : ∀ (F : DLSubPreSheaf) (G : DLPreSheaf) (α : NatTrans (G ∘F (baseIncl ^opF)) F) + → ∃![ σ ∈ NatTrans G (DLRan F) ] α ≡ (σ ∘ˡ (baseIncl ^opF)) ●ᵛ (DLRanNatTrans F) DLRanUnivProp = RanUnivProp _ _ - DLRanNatIso : (F : DLSubPreSheaf) → NatIso (funcComp (DLRan F) (baseIncl ^opF)) F + DLRanNatIso : (F : DLSubPreSheaf) → NatIso (funcComp (DLRan F) (baseIncl ^opF)) F DLRanNatIso F = RanNatIso _ _ _ (λ _ _ → idIsEquiv _) module _ (isBasisL' : IsBasis L L') (F : DLSubPreSheaf) @@ -104,27 +104,27 @@ instance _ = isBasisL' - F* = T* limitC (baseIncl ^opF) F + F* = T* limitC (baseIncl ^opF) F -- a neat lemma F≤PathPLemmaBase : ∀ {x y z w : ob DLSubCat} (p : x ≡ y) (q : z ≡ w) (x≥z : (DLSubCat ^op) [ x , z ]) (y≥w : (DLSubCat ^op) [ y , w ]) - → PathP (λ i → C [ F .F-ob (p i) , F .F-ob (q i) ]) (F .F-hom x≥z) (F .F-hom y≥w) + → PathP (λ i → C [ F .F-ob (p i) , F .F-ob (q i) ]) (F .F-hom x≥z) (F .F-hom y≥w) F≤PathPLemmaBase p q x≥z y≥w i = - F .F-hom (isProp→PathP (λ j → is-prop-valued (q j .fst) (p j .fst)) x≥z y≥w i) + F .F-hom (isProp→PathP (λ j → is-prop-valued (q j .fst) (p j .fst)) x≥z y≥w i) -- the crucial lemmas that will give us the cones needed to construct the unique -- arrow in our pullback square below module _ {n : ℕ} (α : FinVec (fst L) n) (α∈L' : ∀ i → α i ∈ L') where private -- from the definition of the can extension - ⋁α↓ = _↓Diag limitC (baseIncl ^opF) F (⋁ α) + ⋁α↓ = _↓Diag limitC (baseIncl ^opF) F (⋁ α) F[⋁α]Cone = limitC ⋁α↓ (F* (⋁ α)) .limCone -- notation that will be used throughout the file. -- this is the restriction of the limiting cone through which we define -- the Kan-extension to the αᵢ's - restCone : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) (DLRan F .F-ob (⋁ α)) + restCone : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) (DLRan F .F-ob (⋁ α)) coneOut restCone (sing i) = F[⋁α]Cone .coneOut ((α i , α∈L' i) , ind≤⋁ α i) coneOut restCone (pair i j i<j) = F[⋁α]Cone .coneOut ((α i ∧l α j , ∧lClosed _ _ (α∈L' i) (α∈L' j)) @@ -155,71 +155,71 @@ ⋁β∈L' u u∈L' u≤⋁α = subst-∈ L' (β≡ u u≤⋁α) u∈L' βCone : (c : ob C) (u : fst L) (u∈L' : u ∈ L') - → Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c - → Cone (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i))) c + → Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c + → Cone (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i))) c coneOut (βCone c u u∈L' cc) (sing i) = coneOut cc (sing i) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) coneOut (βCone c u u∈L' cc) (pair i j i<j) = coneOut cc (pair i j i<j) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _))) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _))) coneOutCommutes (βCone c u u∈L' cc) {u = v} idAr = cong (λ x → coneOut (βCone c u u∈L' cc) v ⋆⟨ C ⟩ x) - (F-id (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)))) + (F-id (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)))) ∙ ⋆IdR C _ coneOutCommutes (βCone c u u∈L' cc) (singPairL {i = i} {j} {i<j}) = - coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairL) + coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairL) ≡⟨ ⋆Assoc C _ _ _ ⟩ - coneOut cc (sing i) ⋆⟨ C ⟩ (F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairL)) - ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom + coneOut cc (sing i) ⋆⟨ C ⟩ (F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairL)) + ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _) ⋆⟨ DLCat ^op ⟩ - (BDiag (λ i → β u i , β∈L' u u∈L' i) .F-hom (singPairL {i = i} {j} {i<j}))) - ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom + (BDiag (λ i → β u i , β∈L' u u∈L' i) .F-hom (singPairL {i = i} {j} {i<j}))) + ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom {y = β u i ∧l β u j , ∧lClosed _ _ (β∈L' u u∈L' i) (β∈L' u u∈L' j)} x) (is-prop-valued _ _ _ _) ⟩ coneOut cc (sing i) - ⋆⟨ C ⟩ F .F-hom ((BDiag (λ i → α i , α∈L' i)) .F-hom (singPairL {i = i} {j} {i<j}) + ⋆⟨ C ⟩ F .F-hom ((BDiag (λ i → α i , α∈L' i)) .F-hom (singPairL {i = i} {j} {i<j}) ⋆⟨ DLCat ^op ⟩ ≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _))) - ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ + ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ coneOut cc (sing i) - ⋆⟨ C ⟩ ((funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairL {i = i} {j} {i<j})) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) + ⋆⟨ C ⟩ ((funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairL {i = i} {j} {i<j})) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ (coneOut cc (sing i) - ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairL {i = i} {j} {i<j}))) - ⋆⟨ C ⟩ F .F-hom ((≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) - ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom + ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairL {i = i} {j} {i<j}))) + ⋆⟨ C ⟩ F .F-hom ((≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) + ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom {y = β u i ∧l β u j , ∧lClosed _ _ (β∈L' u u∈L' i) (β∈L' u u∈L' j)} (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) (coneOutCommutes cc (singPairL {i = i} {j} {i<j})) ⟩ coneOut (βCone c u u∈L' cc) (pair i j i<j) ∎ coneOutCommutes (βCone c u u∈L' cc) (singPairR {i = i} {j} {i<j}) = - coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairR) + coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairR) ≡⟨ ⋆Assoc C _ _ _ ⟩ - coneOut cc (sing j) ⋆⟨ C ⟩ (F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairR)) - ≡⟨ cong (λ x → coneOut cc (sing j) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom + coneOut cc (sing j) ⋆⟨ C ⟩ (F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → β u i , β∈L' u u∈L' i)) .F-hom singPairR)) + ≡⟨ cong (λ x → coneOut cc (sing j) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _) ⋆⟨ DLCat ^op ⟩ - (BDiag (λ i → β u i , β∈L' u u∈L' i) .F-hom (singPairR {i = i} {j} {i<j}))) - ≡⟨ cong (λ x → coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom + (BDiag (λ i → β u i , β∈L' u u∈L' i) .F-hom (singPairR {i = i} {j} {i<j}))) + ≡⟨ cong (λ x → coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom {y = β u i ∧l β u j , ∧lClosed _ _ (β∈L' u u∈L' i) (β∈L' u u∈L' j)} x) (is-prop-valued _ _ _ _) ⟩ coneOut cc (sing j) - ⋆⟨ C ⟩ F .F-hom ((BDiag (λ i → α i , α∈L' i)) .F-hom (singPairR {i = i} {j} {i<j}) + ⋆⟨ C ⟩ F .F-hom ((BDiag (λ i → α i , α∈L' i)) .F-hom (singPairR {i = i} {j} {i<j}) ⋆⟨ DLCat ^op ⟩ ≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _))) - ≡⟨ cong (λ x → coneOut cc (sing j) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ + ≡⟨ cong (λ x → coneOut cc (sing j) ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ coneOut cc (sing j) - ⋆⟨ C ⟩ ((funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairR {i = i} {j} {i<j})) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) + ⋆⟨ C ⟩ ((funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairR {i = i} {j} {i<j})) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ (coneOut cc (sing j) - ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairR {i = i} {j} {i<j}))) - ⋆⟨ C ⟩ F .F-hom ((≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) - ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom + ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → α i , α∈L' i)) .F-hom (singPairR {i = i} {j} {i<j}))) + ⋆⟨ C ⟩ F .F-hom ((≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) + ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom {y = β u i ∧l β u j , ∧lClosed _ _ (β∈L' u u∈L' i) (β∈L' u u∈L' j)} (≤m→≤j _ _ (≤-∧Pres _ _ _ _ (∧≤LCancel _ _) (∧≤LCancel _ _)))) (coneOutCommutes cc (singPairR {i = i} {j} {i<j})) ⟩ @@ -227,9 +227,9 @@ -- this is the crucial application of our assumption that F is a sheaf on L' - uniqβConeMor : (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) + uniqβConeMor : (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) (u : fst L) (u∈L' : u ∈ L') (u≤⋁α : u ≤ ⋁ α) - → ∃![ f ∈ C [ c , F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) ] ] + → ∃![ f ∈ C [ c , F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) ] ] (isConeMor (βCone c u u∈L' cc) (F-cone F (B⋁Cone (λ i → β u i , β∈L' u u∈L' i) (⋁β∈L' u u∈L' u≤⋁α))) f) uniqβConeMor c cc u u∈L' u≤⋁α = @@ -237,37 +237,37 @@ -- the lemma giving us the desired cone - lemma1 : (c : ob C) → Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c → Cone (F* (⋁ α)) c + lemma1 : (c : ob C) → Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c → Cone (F* (⋁ α)) c coneOut (lemma1 c cc) ((u , u∈L') , u≤⋁α) = - subst (λ x → C [ c , F .F-ob x ]) - (Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' u u∈L' u≤⋁α} (sym (β≡ u u≤⋁α))) + subst (λ x → C [ c , F .F-ob x ]) + (Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' u u∈L' u≤⋁α} (sym (β≡ u u≤⋁α))) (uniqβConeMor c cc u u∈L' u≤⋁α .fst .fst) coneOutCommutes (lemma1 c cc) {u = ((u , u∈L') , u≤⋁α)} {v = ((v , v∈L') , v≤⋁α)} (v≤u , p) = transport (λ i → fᵤPathP i ⋆⟨ C ⟩ ePathP i ≡ fᵥPathP i) triangle where - e : C [ F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) , F .F-ob (⋁ (β v) , ⋁β∈L' v v∈L' v≤⋁α) ] - e = F .F-hom (subst2 _≤_ (β≡ v v≤⋁α) (β≡ u u≤⋁α) v≤u) -- F(⋁βᵥ≤⋁βᵤ) + e : C [ F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) , F .F-ob (⋁ (β v) , ⋁β∈L' v v∈L' v≤⋁α) ] + e = F .F-hom (subst2 _≤_ (β≡ v v≤⋁α) (β≡ u u≤⋁α) v≤u) -- F(⋁βᵥ≤⋁βᵤ) - fᵤ : C [ c , F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) ] + fᵤ : C [ c , F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) ] fᵤ = uniqβConeMor c cc u u∈L' u≤⋁α .fst .fst - fᵥ : C [ c , F .F-ob (⋁ (β v) , ⋁β∈L' v v∈L' v≤⋁α) ] + fᵥ : C [ c , F .F-ob (⋁ (β v) , ⋁β∈L' v v∈L' v≤⋁α) ] fᵥ = uniqβConeMor c cc v v∈L' v≤⋁α .fst .fst -- for convenience - pᵤ = (Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' u u∈L' u≤⋁α} (sym (β≡ u u≤⋁α))) - pᵥ = (Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' v v∈L' v≤⋁α} (sym (β≡ v v≤⋁α))) + pᵤ = (Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' u u∈L' u≤⋁α} (sym (β≡ u u≤⋁α))) + pᵥ = (Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' v v∈L' v≤⋁α} (sym (β≡ v v≤⋁α))) - fᵤPathP : PathP (λ i → C [ c , F .F-ob (pᵤ i) ]) + fᵤPathP : PathP (λ i → C [ c , F .F-ob (pᵤ i) ]) fᵤ (coneOut (lemma1 c cc) ((u , u∈L') , u≤⋁α)) - fᵤPathP = subst-filler (λ x → C [ c , F .F-ob x ]) pᵤ fᵤ + fᵤPathP = subst-filler (λ x → C [ c , F .F-ob x ]) pᵤ fᵤ - fᵥPathP : PathP (λ i → C [ c , F .F-ob (pᵥ i) ]) + fᵥPathP : PathP (λ i → C [ c , F .F-ob (pᵥ i) ]) fᵥ (coneOut (lemma1 c cc) ((v , v∈L') , v≤⋁α)) - fᵥPathP = subst-filler (λ x → C [ c , F .F-ob x ]) pᵥ fᵥ + fᵥPathP = subst-filler (λ x → C [ c , F .F-ob x ]) pᵥ fᵥ - ePathP : PathP (λ i → C [ F .F-ob (pᵤ i) , F .F-ob (pᵥ i) ]) e (F .F-hom v≤u) - ePathP i = F .F-hom (subst2-filler (_≤_) (β≡ v v≤⋁α) (β≡ u u≤⋁α) v≤u (~ i)) + ePathP : PathP (λ i → C [ F .F-ob (pᵤ i) , F .F-ob (pᵥ i) ]) e (F .F-hom v≤u) + ePathP i = F .F-hom (subst2-filler (_≤_) (β≡ v v≤⋁α) (β≡ u u≤⋁α) v≤u (~ i)) -- triangle to be transported by universal property @@ -282,68 +282,68 @@ (F-cone F (B⋁Cone (λ i → β v i , β∈L' v v∈L' i) (⋁β∈L' v v∈L' v≤⋁α))) singCase where - singCase : ∀ i → (fᵤ ⋆⟨ C ⟩ e) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β v) i) - ≡ coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j (v ∧l α i) (α i) (∧≤LCancel _ _)) + singCase : ∀ i → (fᵤ ⋆⟨ C ⟩ e) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β v) i) + ≡ coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j (v ∧l α i) (α i) (∧≤LCancel _ _)) singCase i = - (fᵤ ⋆⟨ C ⟩ e) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β v) i) + (fᵤ ⋆⟨ C ⟩ e) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β v) i) ≡⟨ ⋆Assoc C _ _ _ ⟩ - fᵤ ⋆⟨ C ⟩ (e ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β v) i)) - ≡⟨ cong (λ x → fᵤ ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - fᵤ ⋆⟨ C ⟩ F .F-hom + fᵤ ⋆⟨ C ⟩ (e ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β v) i)) + ≡⟨ cong (λ x → fᵤ ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + fᵤ ⋆⟨ C ⟩ F .F-hom (subst2 _≤_ (β≡ v v≤⋁α) (β≡ u u≤⋁α) v≤u ⋆⟨ DLCat ^op ⟩ ind≤⋁ (β v) i) - ≡⟨ cong (λ x → fᵤ ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ v∈L' (α∈L' i)} x) + ≡⟨ cong (λ x → fᵤ ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ v∈L' (α∈L' i)} x) (is-prop-valued _ _ _ _) ⟩ - fᵤ ⋆⟨ C ⟩ F .F-hom + fᵤ ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β u) i ⋆⟨ DLCat ^op ⟩ ≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u))) - ≡⟨ cong (λ x → fᵤ ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ - fᵤ ⋆⟨ C ⟩ (F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} (ind≤⋁ (β u) i) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u)))) + ≡⟨ cong (λ x → fᵤ ⋆⟨ C ⟩ x) (F .F-seq _ _) ⟩ + fᵤ ⋆⟨ C ⟩ (F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} (ind≤⋁ (β u) i) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u)))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - (fᵤ ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} (ind≤⋁ (β u) i)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u))) - ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u)))) + (fᵤ ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} (ind≤⋁ (β u) i)) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u))) + ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u)))) (uniqβConeMor c cc u u∈L' u≤⋁α .fst .snd (sing i)) ⟩ - (coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} + (coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} (≤m→≤j (u ∧l α i) (α i) (∧≤LCancel _ _))) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u))) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u))) ≡⟨ ⋆Assoc C _ _ _ ⟩ - coneOut cc (sing i) ⋆⟨ C ⟩ (F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} + coneOut cc (sing i) ⋆⟨ C ⟩ (F .F-hom {y = _ , ∧lClosed _ _ u∈L' (α∈L' i)} (≤m→≤j (u ∧l α i) (α i) (∧≤LCancel _ _)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u)))) - ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ - coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u)))) + ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ x) (sym (F .F-seq _ _)) ⟩ + coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j (u ∧l α i) (α i) (∧≤LCancel _ _) ⋆⟨ DLCat ^op ⟩ ≤m→≤j _ _ (≤-∧RPres _ _ _ (≤j→≤m _ _ v≤u))) - ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom + ≡⟨ cong (λ x → coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ v∈L' (α∈L' i)} x) (is-prop-valued _ _ _ _) ⟩ - coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j (v ∧l α i) (α i) (∧≤LCancel _ _)) ∎ + coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j (v ∧l α i) (α i) (∧≤LCancel _ _)) ∎ -- gives us preservation of cone morphisms that ensure uniqueness - lemma2 : ∀ (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) - (f : C [ c , (DLRan F .F-ob (⋁ α)) ]) + lemma2 : ∀ (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) + (f : C [ c , (DLRan F .F-ob (⋁ α)) ]) → isConeMor cc restCone f → isConeMor (lemma1 c cc) F[⋁α]Cone f lemma2 c cc f isRestConeMorf ((u , u∈L') , u≤⋁α) = transport (λ i → f ⋆⟨ C ⟩ coneOutPathP i ≡ bᵤPathP i) triangle where -- for convenience - pᵤ = Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' u u∈L' u≤⋁α} + pᵤ = Σ≡Prop (λ x → L' x .snd) {u = _ , ⋁β∈L' u u∈L' u≤⋁α} {v = _ , u∈L'} (sym (β≡ u u≤⋁α)) - bᵤ : C [ c , F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) ] + bᵤ : C [ c , F .F-ob (⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) ] bᵤ = uniqβConeMor c cc u u∈L' u≤⋁α .fst .fst - bᵤPathP : PathP (λ i → C [ c , F .F-ob (pᵤ i) ]) + bᵤPathP : PathP (λ i → C [ c , F .F-ob (pᵤ i) ]) bᵤ (coneOut (lemma1 c cc) ((u , u∈L') , u≤⋁α)) - bᵤPathP = subst-filler (λ x → C [ c , F .F-ob x ]) pᵤ bᵤ + bᵤPathP = subst-filler (λ x → C [ c , F .F-ob x ]) pᵤ bᵤ ⋁βᵤ : ob ⋁α↓ ⋁βᵤ = ((⋁ (β u) , ⋁β∈L' u u∈L' u≤⋁α) , subst (_≤ ⋁ α) (β≡ u u≤⋁α) u≤⋁α) - coneOutPathP : PathP (λ i → C [ (DLRan F .F-ob (⋁ α)) , F .F-ob (pᵤ i) ]) + coneOutPathP : PathP (λ i → C [ (DLRan F .F-ob (⋁ α)) , F .F-ob (pᵤ i) ]) (coneOut F[⋁α]Cone ⋁βᵤ) (coneOut F[⋁α]Cone ((u , u∈L') , u≤⋁α)) coneOutPathP i = coneOut F[⋁α]Cone ((pᵤ i) , subst-filler (_≤ ⋁ α) (β≡ u u≤⋁α) u≤⋁α (~ i)) @@ -362,28 +362,28 @@ u∧αᵢ≤⋁α : ∀ i → (DLCat ^op) [ ⋁ α , u ∧l α i ] u∧αᵢ≤⋁α _ = is-trans _ _ _ (≤m→≤j _ _ (∧≤RCancel _ _)) u≤⋁α - singCase : ∀ i → (f ⋆⟨ C ⟩ coneOut F[⋁α]Cone ⋁βᵤ) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β u) i) + singCase : ∀ i → (f ⋆⟨ C ⟩ coneOut F[⋁α]Cone ⋁βᵤ) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β u) i) ≡ coneOut (βCone c u u∈L' cc) (sing i) singCase i = - (f ⋆⟨ C ⟩ coneOut F[⋁α]Cone ⋁βᵤ) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β u) i) + (f ⋆⟨ C ⟩ coneOut F[⋁α]Cone ⋁βᵤ) ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β u) i) ≡⟨ ⋆Assoc C _ _ _ ⟩ - f ⋆⟨ C ⟩ (coneOut F[⋁α]Cone ⋁βᵤ ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β u) i)) + f ⋆⟨ C ⟩ (coneOut F[⋁α]Cone ⋁βᵤ ⋆⟨ C ⟩ F .F-hom (ind≤⋁ (β u) i)) ≡⟨ cong (λ x → f ⋆⟨ C ⟩ x) (coneOutCommutes F[⋁α]Cone (ind≤⋁ (β u) i , is-prop-valued _ _ _ _)) ⟩ f ⋆⟨ C ⟩ coneOut F[⋁α]Cone ((u ∧l α i , ∧lClosed _ _ u∈L' (α∈L' i)) , u∧αᵢ≤⋁α i) ≡⟨ cong (λ x → f ⋆⟨ C ⟩ x) (sym (coneOutCommutes F[⋁α]Cone (≤m→≤j _ _ (∧≤LCancel _ _) , is-prop-valued _ _ _ _))) ⟩ f ⋆⟨ C ⟩ (coneOut F[⋁α]Cone ((α i , α∈L' i) , ind≤⋁ α i) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ (f ⋆⟨ C ⟩ coneOut F[⋁α]Cone ((α i , α∈L' i) , ind≤⋁ α i)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) (isRestConeMorf (sing i)) ⟩ - coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ∎ + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ≡⟨ cong (λ x → x ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) (isRestConeMorf (sing i)) ⟩ + coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ∎ -- the other direction, surprisingly hard - lemma3 : ∀ (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) - (f : C [ c , DLRan F .F-ob (⋁ α) ]) + lemma3 : ∀ (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) + (f : C [ c , DLRan F .F-ob (⋁ α) ]) → isConeMor (lemma1 c cc) F[⋁α]Cone f → isConeMor cc restCone f lemma3 c cc f isConeMorF = isConeMorSingLemma cc restCone singCase @@ -400,24 +400,24 @@ -- modulo transport Σpathhelper : (α i , α∈L' i) ≡ (⋁ (β (α i)) , ⋁β∈L' (α i) (α∈L' i) (ind≤⋁ α i)) - Σpathhelper = Σ≡Prop (λ x → L' x .snd) (β≡ (α i) (ind≤⋁ α i)) + Σpathhelper = Σ≡Prop (λ x → L' x .snd) (β≡ (α i) (ind≤⋁ α i)) Σpathhelper2 : (⋁ (β (α i)) , ⋁β∈L' (α i) (α∈L' i) (ind≤⋁ α i)) ≡ (α i , α∈L' i) - Σpathhelper2 = Σ≡Prop (λ x → L' x .snd) (sym (β≡ (α i) (ind≤⋁ α i))) + Σpathhelper2 = Σ≡Prop (λ x → L' x .snd) (sym (β≡ (α i) (ind≤⋁ α i))) - ccᵢSubst : C [ c , F .F-ob (⋁ (β (α i)) , ⋁β∈L' (α i) (α∈L' i) (ind≤⋁ α i)) ] - ccᵢSubst = subst (λ x → C [ c , F .F-ob x ]) - (Σ≡Prop (λ x → L' x .snd) (β≡ (α i) (ind≤⋁ α i))) + ccᵢSubst : C [ c , F .F-ob (⋁ (β (α i)) , ⋁β∈L' (α i) (α∈L' i) (ind≤⋁ α i)) ] + ccᵢSubst = subst (λ x → C [ c , F .F-ob x ]) + (Σ≡Prop (λ x → L' x .snd) (β≡ (α i) (ind≤⋁ α i))) (coneOut cc (sing i)) - ccᵢSubstFiller : PathP (λ j → C [ c , F .F-ob (Σpathhelper j) ]) + ccᵢSubstFiller : PathP (λ j → C [ c , F .F-ob (Σpathhelper j) ]) (coneOut cc (sing i)) ccᵢSubst - ccᵢSubstFiller = subst-filler (λ x → C [ c , F .F-ob x ]) Σpathhelper (coneOut cc (sing i)) + ccᵢSubstFiller = subst-filler (λ x → C [ c , F .F-ob x ]) Σpathhelper (coneOut cc (sing i)) - βSubstFiller : PathP (λ j → C [ c , F .F-ob (Σpathhelper2 j) ]) + βSubstFiller : PathP (λ j → C [ c , F .F-ob (Σpathhelper2 j) ]) (uniqβConeMor c cc (α i) (α∈L' i) (ind≤⋁ α i) .fst .fst) (coneOut (lemma1 c cc) ((α i , α∈L' i) , ind≤⋁ α i)) - βSubstFiller = subst-filler (λ x → C [ c , F .F-ob x ]) Σpathhelper2 + βSubstFiller = subst-filler (λ x → C [ c , F .F-ob x ]) Σpathhelper2 (uniqβConeMor c cc (α i) (α∈L' i) (ind≤⋁ α i) .fst .fst) Σpathhelperpath : Σpathhelper2 ≡ sym Σpathhelper @@ -426,10 +426,10 @@ isSetL' : isSet (ob DLSubCat) isSetL' = isSetΣSndProp is-set λ x → L' x .snd - helperPathP : PathP (λ j → C [ c , F .F-ob (Σpathhelper (~ j)) ]) + helperPathP : PathP (λ j → C [ c , F .F-ob (Σpathhelper (~ j)) ]) (uniqβConeMor c cc (α i) (α∈L' i) (ind≤⋁ α i) .fst .fst) (coneOut (lemma1 c cc) ((α i , α∈L' i) , ind≤⋁ α i)) - helperPathP = subst (λ p → PathP (λ j → C [ c , F .F-ob (p j) ]) + helperPathP = subst (λ p → PathP (λ j → C [ c , F .F-ob (p j) ]) (uniqβConeMor c cc (α i) (α∈L' i) (ind≤⋁ α i) .fst .fst) (coneOut (lemma1 c cc) ((α i , α∈L' i) , ind≤⋁ α i))) Σpathhelperpath βSubstFiller @@ -443,8 +443,8 @@ (⋁β∈L' (α i) (α∈L' i) (ind≤⋁ α i)))) singCase2 where - singCase2 : (j : Fin n) → ccᵢSubst ⋆⟨ C ⟩ F-hom F (ind≤⋁ (β (α i)) j) - ≡ coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + singCase2 : (j : Fin n) → ccᵢSubst ⋆⟨ C ⟩ F-hom F (ind≤⋁ (β (α i)) j) + ≡ coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) singCase2 j = (λ 𝕚 → ccᵢSubstFiller (~ 𝕚) ⋆⟨ C ⟩ F≤PathPLemmaBase @@ -452,8 +452,8 @@ (ind≤⋁ (β (α i)) j) (≤m→≤j _ _ (∧≤RCancel _ _)) 𝕚) ∙ path where - path : coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) - ≡ coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + path : coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + ≡ coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) path with (i ≟Fin j) ... | (lt i<j) = coneOutCommutes cc (singPairL {i<j = i<j}) ∙ sym (coneOutCommutes cc singPairR) @@ -461,10 +461,10 @@ where ∧Path : Path (ob DLSubCat) (α j ∧l α i , β∈L' (α j) (α∈L' j) i) (α i ∧l α j , β∈L' (α i) (α∈L' i) j) - ∧Path = Σ≡Prop (λ x → L' x .snd) (∧lComm _ _) + ∧Path = Σ≡Prop (λ x → L' x .snd) (∧lComm _ _) - almostPath : coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - ≡ coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + almostPath : coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ≡ coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) almostPath = (coneOutCommutes cc (singPairR {i<j = j<i}) ∙ sym (coneOutCommutes cc singPairL)) @@ -477,13 +477,13 @@ (≤m→≤j _ _ (∧≤LCancel _ _)) 𝕚 ... | (eq i≡j) = - coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + coneOut cc (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ≡⟨ (λ 𝕚 → coneOut cc (sing (i≡j 𝕚)) ⋆⟨ C ⟩ F≤PathPLemmaBase (λ 𝕛 → α (i≡j 𝕛) , α∈L' (i≡j 𝕛)) refl (≤m→≤j _ _ (∧≤RCancel _ _)) (≤m→≤j _ _ (∧≤LCancel _ _)) 𝕚) ⟩ - coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ∎ + coneOut cc (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ∎ ccᵢSubstPath : uniqβConeMor c cc (α i) (α∈L' i) (ind≤⋁ α i) .fst .fst ≡ ccᵢSubst @@ -493,15 +493,15 @@ -- putting it all together - coverLemma : ∀ (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) - → ∃![ f ∈ C [ c , DLRan F .F-ob (⋁ α) ] ] isConeMor cc restCone f + coverLemma : ∀ (c : ob C) (cc : Cone (funcComp F (BDiag (λ i → α i , α∈L' i))) c) + → ∃![ f ∈ C [ c , DLRan F .F-ob (⋁ α) ] ] isConeMor cc restCone f coverLemma c cc = uniqueExists (fromUnivProp .fst .fst) (lemma3 c cc _ (fromUnivProp .fst .snd)) (λ _ → isPropIsConeMor _ _ _) λ g isConeMorG → cong fst (fromUnivProp .snd (g , lemma2 c cc g isConeMorG)) where - fromUnivProp : ∃![ f ∈ C [ c , DLRan F .F-ob (⋁ α) ] ] isConeMor (lemma1 c cc) F[⋁α]Cone f + fromUnivProp : ∃![ f ∈ C [ c , DLRan F .F-ob (⋁ α) ] ] isConeMor (lemma1 c cc) F[⋁α]Cone f fromUnivProp = limitC ⋁α↓ (F* (⋁ α)) .univProp c (lemma1 c cc) @@ -531,7 +531,7 @@ ++FinInrΣ {ℕ.suc n} β∈L' γ∈L' i = ++FinInrΣ (β∈L' ∘ suc) γ∈L' i module ++Lemmas {c : ob C} {n' : ℕ} {γ : FinVec (fst L) n'} {γ∈L' : ∀ i → γ i ∈ L'} - (ccγ : Cone (funcComp F (BDiag (λ i → γ i , γ∈L' i))) c) where + (ccγ : Cone (funcComp F (BDiag (λ i → γ i , γ∈L' i))) c) where private β∧γ : {n : ℕ} {β : FinVec (fst L) n} (β∈L' : ∀ i → β i ∈ L') @@ -547,16 +547,16 @@ γ≥β∧γ β∈L' i j = ≤m→≤j _ _ (∧≤LCancel _ _) CommHypType : {n : ℕ} {β : FinVec (fst L) n} (β∈L' : ∀ i → β i ∈ L') - (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) + (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) → Type ℓ'' CommHypType β∈L' ccβ = ∀ i j → ccβ .coneOut (sing i) - ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ (β∈L' i) (γ∈L' j)} (β≥β∧γ β∈L' i j) - ≡ ccγ .coneOut (sing j) ⋆⟨ C ⟩ F .F-hom (γ≥β∧γ β∈L' i j) + ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ (β∈L' i) (γ∈L' j)} (β≥β∧γ β∈L' i j) + ≡ ccγ .coneOut (sing j) ⋆⟨ C ⟩ F .F-hom (γ≥β∧γ β∈L' i j) coneSuc : {n : ℕ} {β : FinVec (fst L) (ℕ.suc n)} {β∈L' : ∀ i → β i ∈ L'} - → Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c - → Cone (funcComp F (BDiag (λ i → β (suc i) , β∈L' (suc i)))) c + → Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c + → Cone (funcComp F (BDiag (λ i → β (suc i) , β∈L' (suc i)))) c coneOut (coneSuc ccβ) (sing i) = coneOut ccβ (sing (suc i)) coneOut (coneSuc ccβ) (pair i j i<j) = coneOut ccβ (pair (suc i) (suc j) (s≤s i<j)) coneOutCommutes (coneSuc ccβ) {u = sing i} idAr = coneOutCommutes ccβ idAr @@ -566,23 +566,23 @@ --make this explicit to avoid yellow commHypSuc : {n : ℕ} {β : FinVec (fst L) (ℕ.suc n)} {β∈L' : ∀ i → β i ∈ L'} - {ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c} + {ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c} → CommHypType β∈L' ccβ → CommHypType (β∈L' ∘ suc) (coneSuc ccβ) commHypSuc commHyp i j = commHyp (suc i) j toConeOut : (n : ℕ) (β : FinVec (fst L) n) (β∈L' : ∀ i → β i ∈ L') - (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) + (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) (ch : CommHypType β∈L' ccβ) → ∀ (v : DLShfDiagOb (n + n')) - → C [ c , funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i)) .F-ob v ] + → C [ c , funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i)) .F-ob v ] toConeOut ℕ.zero β β∈L' ccβ ch (sing i) = ccγ .coneOut (sing i) toConeOut ℕ.zero β β∈L' ccβ ch (pair i j i<j) = ccγ .coneOut (pair i j i<j) toConeOut (ℕ.suc n) β β∈L' ccβ ch (sing zero) = ccβ .coneOut (sing zero) toConeOut (ℕ.suc n) β β∈L' ccβ ch (sing (suc i)) = toConeOut n (β ∘ suc) (β∈L' ∘ suc) (coneSuc ccβ) (commHypSuc ch) (sing i) toConeOut (ℕ.suc n) β β∈L' ccβ ch (pair zero j 0<j) = - ccβ .coneOut (sing zero) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + ccβ .coneOut (sing zero) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) toConeOut (ℕ.suc n) β β∈L' ccβ ch (pair (suc i) zero ()) toConeOut (ℕ.suc n) β β∈L' ccβ ch (pair (suc i) (suc j) (s≤s i<j)) = toConeOut n (β ∘ suc) (β∈L' ∘ suc) (coneSuc ccβ) (commHypSuc ch) (pair i j i<j) @@ -590,23 +590,23 @@ -- crucial step in proving that this defines a cone is another induction -- βₛ is supposed to be (β ∘ suc) and β₀ is (β zero) toConeOutLemma : (n : ℕ) (βₛ : FinVec (fst L) n) (βₛ∈L' : ∀ i → βₛ i ∈ L') - (ccβₛ : Cone (funcComp F (BDiag (λ i → βₛ i , βₛ∈L' i))) c) + (ccβₛ : Cone (funcComp F (BDiag (λ i → βₛ i , βₛ∈L' i))) c) (chₛ : CommHypType βₛ∈L' ccβₛ) (β₀ : fst L) (β₀∈L' : β₀ ∈ L') -- cone over [β₀]++βₛ - {ccβ₀ : C [ c , F .F-ob (β₀ , β₀∈L') ]} - {ccβ₀ᵢ : (i : Fin n) → C [ c , F .F-ob (β₀ ∧l βₛ i , ∧lClosed _ _ β₀∈L' (βₛ∈L' i)) ]} - (ccβ₀L : ∀ i → ccβ₀ ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ≡ ccβ₀ᵢ i) - (ccβ₀R : ∀ i → ccβₛ .coneOut (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ≡ ccβ₀ᵢ i) + {ccβ₀ : C [ c , F .F-ob (β₀ , β₀∈L') ]} + {ccβ₀ᵢ : (i : Fin n) → C [ c , F .F-ob (β₀ ∧l βₛ i , ∧lClosed _ _ β₀∈L' (βₛ∈L' i)) ]} + (ccβ₀L : ∀ i → ccβ₀ ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ≡ ccβ₀ᵢ i) + (ccβ₀R : ∀ i → ccβₛ .coneOut (sing i) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ≡ ccβ₀ᵢ i) -- ch at zero (ch₀ : ∀ j → - ccβ₀ ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ β₀∈L' (γ∈L' j)} (≤m→≤j _ _ (∧≤RCancel _ _)) - ≡ ccγ .coneOut (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) + ccβ₀ ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ β₀∈L' (γ∈L' j)} (≤m→≤j _ _ (∧≤RCancel _ _)) + ≡ ccγ .coneOut (sing j) ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) --------------------------------------------------------------------- → ∀ j → toConeOut n βₛ βₛ∈L' ccβₛ chₛ (sing j) - ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ β₀∈L' (β++γ∈L' βₛ∈L' γ∈L' j)} + ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ β₀∈L' (β++γ∈L' βₛ∈L' γ∈L' j)} (≤m→≤j _ _ (∧≤LCancel _ _)) - ≡ ccβ₀ ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + ≡ ccβ₀ ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) toConeOutLemma ℕ.zero _ _ _ _ _ _ _ _ ch₀ j = sym (ch₀ j) toConeOutLemma (ℕ.suc n) _ _ _ _ _ _ ccβ₀L ccβ₀R _ zero = ccβ₀R zero ∙ sym (ccβ₀L zero) toConeOutLemma (ℕ.suc n) βₛ βₛ∈L' ccβₛ chₛ β₀ β₀∈L' ccβ₀L ccβ₀R ch₀ (suc j) = @@ -615,11 +615,11 @@ toConeOutCommutes : (n : ℕ) (β : FinVec (fst L) n) (β∈L' : ∀ i → β i ∈ L') - (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) + (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) (ch : CommHypType β∈L' ccβ) → ∀ {u} {v} e → toConeOut _ _ _ ccβ ch u - ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i)) .F-hom e) + ⋆⟨ C ⟩ (funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i)) .F-hom e) ≡ toConeOut _ _ _ ccβ ch v toConeOutCommutes ℕ.zero _ _ _ _ {u = sing i} {v = sing .i} idAr = coneOutCommutes ccγ idAr toConeOutCommutes ℕ.zero _ _ _ _ {u = sing i} {v = pair .i j i<j} singPairL = @@ -630,7 +630,7 @@ toConeOutCommutes ℕ.zero _ _ _ _ {u = pair i j i<j} {v = pair .i .j .i<j} idAr = coneOutCommutes ccγ idAr toConeOutCommutes (ℕ.suc n) β β∈L' ccβ ch idAr = - cong (λ x → toConeOut _ _ _ ccβ ch _ ⋆⟨ C ⟩ x) (F .F-id) ∙ ⋆IdR C _ + cong (λ x → toConeOut _ _ _ ccβ ch _ ⋆⟨ C ⟩ x) (F .F-id) ∙ ⋆IdR C _ toConeOutCommutes (ℕ.suc n) β β∈L' ccβ ch (singPairL {i = zero} {j = j} {i<j = i<j}) = refl toConeOutCommutes (ℕ.suc n) β β∈L' ccβ ch (singPairL {i = suc i} {j = zero} {i<j = ()}) toConeOutCommutes (ℕ.suc n) β β∈L' ccβ ch (singPairL {i = suc i} {j = suc j} {i<j = s≤s i<j}) = @@ -644,18 +644,18 @@ (ch zero) j toCone : {n : ℕ} {β : FinVec (fst L) n} {β∈L' : ∀ i → β i ∈ L'} - (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) + (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) (ch : CommHypType β∈L' ccβ) - → Cone (funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i))) c + → Cone (funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i))) c coneOut (toCone ccβ ch) = toConeOut _ _ _ ccβ ch coneOutCommutes (toCone ccβ ch) = toConeOutCommutes _ _ _ ccβ ch -- for checking the universal property toConeOutPathPL : {n : ℕ} {β : FinVec (fst L) n} {β∈L' : ∀ i → β i ∈ L'} - (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) + (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) (ch : CommHypType β∈L' ccβ) - → ∀ i → PathP (λ 𝕚 → C [ c , F .F-ob (++FinInlΣ β∈L' γ∈L' i 𝕚) ]) + → ∀ i → PathP (λ 𝕚 → C [ c , F .F-ob (++FinInlΣ β∈L' γ∈L' i 𝕚) ]) (ccβ .coneOut (sing i)) (toCone ccβ ch .coneOut (sing (FSCfun _ _ (inl i)))) toConeOutPathPL {ℕ.zero} ccβ _ () @@ -663,9 +663,9 @@ toConeOutPathPL {ℕ.suc n} ccβ ch (suc i) = toConeOutPathPL (coneSuc ccβ) (commHypSuc ch) i toConeOutPathPR : {n : ℕ} {β : FinVec (fst L) n} {β∈L' : ∀ i → β i ∈ L'} - (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) + (ccβ : Cone (funcComp F (BDiag (λ i → β i , β∈L' i))) c) (ch : CommHypType β∈L' ccβ) - → ∀ i → PathP (λ 𝕚 → C [ c , F .F-ob (++FinInrΣ β∈L' γ∈L' i 𝕚) ]) + → ∀ i → PathP (λ 𝕚 → C [ c , F .F-ob (++FinInrΣ β∈L' γ∈L' i 𝕚) ]) (ccγ .coneOut (sing i)) (toCone ccβ ch .coneOut (sing (FSCfun _ _ (inr i)))) toConeOutPathPR {ℕ.zero} ccβ _ i = refl @@ -678,11 +678,11 @@ limArrow (limitC _ (F* 0l)) x (toCone x) , λ f → limArrowUnique (limitC _ (F* 0l)) x (toCone x) f (toConeMor x f) where - 0↓ = _↓Diag limitC (baseIncl ^opF) F 0l + 0↓ = _↓Diag limitC (baseIncl ^opF) F 0l - toTerminal : ∀ (u : ob 0↓) → isTerminal C (F .F-ob (u .fst)) - toTerminal ((u , u∈L') , 0≥u) = subst (λ v → isTerminal C (F .F-ob v)) - (Σ≡Prop (λ y → L' y .snd) 0≡u) + toTerminal : ∀ (u : ob 0↓) → isTerminal C (F .F-ob (u .fst)) + toTerminal ((u , u∈L') , 0≥u) = subst (λ v → isTerminal C (F .F-ob v)) + (Σ≡Prop (λ y → L' y .snd) 0≡u) (DLBasisSheaf→Terminal F isSheafF 0∈L') where 0≡u : 0l ≡ u @@ -694,7 +694,7 @@ coneOut (toCone y) u = toTerminal u y .fst coneOutCommutes (toCone y) {v = v} e = sym (toTerminal v y .snd _) - toConeMor : (y : ob C) (f : C [ y , F-ob (DLRan F) 0l ]) + toConeMor : (y : ob C) (f : C [ y , F-ob (DLRan F) 0l ]) → isConeMor (toCone y) (limCone (limitC 0↓ (F* 0l))) f toConeMor y f v = sym (toTerminal v y .snd _) @@ -717,25 +717,25 @@ -- replace x and y by their representations of joins of base elements -- and transport over xyCospan : Cospan C - l xyCospan = DLRan F .F-ob y - m xyCospan = DLRan F .F-ob (x ∧l y) - r xyCospan = DLRan F .F-ob x - s₁ xyCospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - s₂ xyCospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + l xyCospan = DLRan F .F-ob y + m xyCospan = DLRan F .F-ob (x ∧l y) + r xyCospan = DLRan F .F-ob x + s₁ xyCospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + s₂ xyCospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ⋁Cospan : Cospan C - l ⋁Cospan = DLRan F .F-ob (⋁ γ) - m ⋁Cospan = DLRan F .F-ob (⋁ β ∧l ⋁ γ) - r ⋁Cospan = DLRan F .F-ob (⋁ β) - s₁ ⋁Cospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - s₂ ⋁Cospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + l ⋁Cospan = DLRan F .F-ob (⋁ γ) + m ⋁Cospan = DLRan F .F-ob (⋁ β ∧l ⋁ γ) + r ⋁Cospan = DLRan F .F-ob (⋁ β) + s₁ ⋁Cospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + s₂ ⋁Cospan = DLRan F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) cospanPath : ⋁Cospan ≡ xyCospan - l (cospanPath i) = DLRan F .F-ob (⋁γ≡y i) - m (cospanPath i) = DLRan F .F-ob (⋁β≡x i ∧l ⋁γ≡y i) - r (cospanPath i) = DLRan F .F-ob (⋁β≡x i) - s₁ (cospanPath i) = DLRan F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) - s₂ (cospanPath i) = DLRan F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + l (cospanPath i) = DLRan F .F-ob (⋁γ≡y i) + m (cospanPath i) = DLRan F .F-ob (⋁β≡x i ∧l ⋁γ≡y i) + r (cospanPath i) = DLRan F .F-ob (⋁β≡x i) + s₁ (cospanPath i) = DLRan F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + s₂ (cospanPath i) = DLRan F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) private F[⋁β]Cone = limitC _ (F* (⋁ β)) .limCone @@ -748,16 +748,16 @@ → f ⋆⟨ C ⟩ ⋁Cospan .s₁ ≡ g ⋆⟨ C ⟩ ⋁Cospan .s₂ → ∀ i j → (g ⋆⟨ C ⟩ restCone β β∈L' .coneOut (sing i)) - ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ (β∈L' i) (γ∈L' j)} + ⋆⟨ C ⟩ F .F-hom {y = _ , ∧lClosed _ _ (β∈L' i) (γ∈L' j)} (≤m→≤j _ _ (∧≤RCancel _ _)) ≡ (f ⋆⟨ C ⟩ restCone γ γ∈L' .coneOut (sing j)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) to++ConeSquare f g square i j = (g ⋆⟨ C ⟩ restCone β β∈L' .coneOut (sing i)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _)) ≡⟨ ⋆Assoc C _ _ _ ⟩ g ⋆⟨ C ⟩ (restCone β β∈L' .coneOut (sing i) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _))) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤RCancel _ _))) ≡⟨ cong (λ x → g ⋆⟨ C ⟩ x) (coneOutCommutes F[⋁β]Cone (_ , (is-prop-valued _ _ _ _))) ⟩ g ⋆⟨ C ⟩ coneOut F[⋁β]Cone ((β i ∧l γ j , _) , is-trans _ _ _ (≤m→≤j _ _ (≤-∧Pres _ _ _ _ @@ -786,18 +786,18 @@ ≡⟨ cong (λ x → f ⋆⟨ C ⟩ x) (sym (coneOutCommutes F[⋁γ]Cone (_ , (is-prop-valued _ _ _ _)))) ⟩ f ⋆⟨ C ⟩ (restCone γ γ∈L' .coneOut (sing j) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _))) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ (f ⋆⟨ C ⟩ restCone γ γ∈L' .coneOut (sing j)) - ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ∎ + ⋆⟨ C ⟩ F .F-hom (≤m→≤j _ _ (∧≤LCancel _ _)) ∎ -- the pullback square we want ⋁Pullback : Pullback C ⋁Cospan - pbOb ⋁Pullback = DLRan F .F-ob (⋁ (β ++Fin γ)) - pbPr₁ ⋁Pullback = DLRan F .F-hom (subst (⋁ γ ≤_) (sym (⋁Split++ β γ)) (∨≤LCancel _ _)) - pbPr₂ ⋁Pullback = DLRan F .F-hom (subst (⋁ β ≤_) (sym (⋁Split++ β γ)) (∨≤RCancel _ _)) - pbCommutes ⋁Pullback = F-square (DLRan F) (is-prop-valued _ _ _ _) + pbOb ⋁Pullback = DLRan F .F-ob (⋁ (β ++Fin γ)) + pbPr₁ ⋁Pullback = DLRan F .F-hom (subst (⋁ γ ≤_) (sym (⋁Split++ β γ)) (∨≤LCancel _ _)) + pbPr₂ ⋁Pullback = DLRan F .F-hom (subst (⋁ β ≤_) (sym (⋁Split++ β γ)) (∨≤RCancel _ _)) + pbCommutes ⋁Pullback = F-square (DLRan F) (is-prop-valued _ _ _ _) univProp ⋁Pullback {d = c} f g square = uniqueExists (applyCoverLemma f g square .fst .fst) (fromConeMor _ (applyCoverLemma f g square .fst .snd)) @@ -808,7 +808,7 @@ toCone : (f : C [ c , ⋁Cospan .l ]) (g : C [ c , ⋁Cospan .r ]) → f ⋆⟨ C ⟩ ⋁Cospan .s₁ ≡ g ⋆⟨ C ⟩ ⋁Cospan .s₂ - → Cone (funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i))) c + → Cone (funcComp F (BDiag (λ i → (β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i))) c toCone f g square = ++Lemmas.toCone (f ★ (restCone γ γ∈L')) (g ★ (restCone β β∈L')) (to++ConeSquare f g square) @@ -821,8 +821,8 @@ -- Another description of the limiting cone over β++γ that -- turns out equivalent but behaves better with the ++Lemmas - ++LimCone' : Cone (funcComp F (BDiag (λ i → ((β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i)))) - (DLRan F .F-ob (⋁ (β ++Fin γ))) + ++LimCone' : Cone (funcComp F (BDiag (λ i → ((β ++Fin γ) i , β++γ∈L' β∈L' γ∈L' i)))) + (DLRan F .F-ob (⋁ (β ++Fin γ))) ++LimCone' = ++Lemmas.toCone ((pbPr₁ ⋁Pullback) ★ (restCone γ γ∈L')) ((pbPr₂ ⋁Pullback) ★ (restCone β β∈L')) (to++ConeSquare _ _ (pbCommutes ⋁Pullback)) @@ -840,8 +840,8 @@ ((pbPr₁ ⋁Pullback) ★ (restCone γ γ∈L')) ((pbPr₂ ⋁Pullback) ★ (restCone β β∈L')) (to++ConeSquare _ _ (pbCommutes ⋁Pullback)) i)) - ∙∙ cong (λ x → transport (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , - F .F-ob (++FinInlΣ β∈L' γ∈L' i 𝕚) ]) x) + ∙∙ cong (λ x → transport (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , + F .F-ob (++FinInlΣ β∈L' γ∈L' i 𝕚) ]) x) (limArrowCommutes (limitC _ (F* (⋁ β))) _ _ _) ∙∙ fromPathP helperPathP where @@ -849,7 +849,7 @@ is-trans _ _ _ (ind≤⋁ β i) (subst (⋁ β ≤_) (sym (⋁Split++ β γ)) (∨≤RCancel _ _)) helperPathP : - PathP (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , F .F-ob (++FinInlΣ β∈L' γ∈L' i 𝕚) ]) + PathP (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , F .F-ob (++FinInlΣ β∈L' γ∈L' i 𝕚) ]) (F[⋁β++γ]Cone .coneOut ((β i , β∈L' i) , βᵢ≤⋁β++γ)) (restCone (β ++Fin γ) (β++γ∈L' β∈L' γ∈L') .coneOut (sing (FSCfun _ _ (inl i)))) helperPathP 𝕚 = F[⋁β++γ]Cone .coneOut (++FinInlΣ β∈L' γ∈L' i 𝕚 , @@ -862,8 +862,8 @@ ((pbPr₁ ⋁Pullback) ★ (restCone γ γ∈L')) ((pbPr₂ ⋁Pullback) ★ (restCone β β∈L')) (to++ConeSquare _ _ (pbCommutes ⋁Pullback)) i)) - ∙∙ cong (λ x → transport (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , - F .F-ob (++FinInrΣ β∈L' γ∈L' i 𝕚) ]) x) + ∙∙ cong (λ x → transport (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , + F .F-ob (++FinInrΣ β∈L' γ∈L' i 𝕚) ]) x) (limArrowCommutes (limitC _ (F* (⋁ γ))) _ _ _) ∙∙ fromPathP helperPathP where @@ -871,7 +871,7 @@ is-trans _ _ _ (ind≤⋁ γ i) (subst (⋁ γ ≤_) (sym (⋁Split++ β γ)) (∨≤LCancel _ _)) helperPathP : - PathP (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , F .F-ob (++FinInrΣ β∈L' γ∈L' i 𝕚) ]) + PathP (λ 𝕚 → C [ DLRan F .F-ob (⋁ (β ++Fin γ)) , F .F-ob (++FinInrΣ β∈L' γ∈L' i 𝕚) ]) (F[⋁β++γ]Cone .coneOut ((γ i , γ∈L' i) , γᵢ≤⋁β++γ)) (restCone (β ++Fin γ) (β++γ∈L' β∈L' γ∈L') .coneOut (sing (FSCfun _ _ (inr i)))) helperPathP 𝕚 = F[⋁β++γ]Cone .coneOut (++FinInrΣ β∈L' γ∈L' i 𝕚 , @@ -1008,14 +1008,14 @@ -- some more names to make the transport readable - pbPr₁PathP : PathP (λ i → C [ DLRan F .F-ob (⋁β++γ≡x∨y i) , DLRan F .F-ob (⋁γ≡y i) ]) - (pbPr₁ ⋁Pullback) (DLRan F .F-hom (hom-∨₂ L C x y)) + pbPr₁PathP : PathP (λ i → C [ DLRan F .F-ob (⋁β++γ≡x∨y i) , DLRan F .F-ob (⋁γ≡y i) ]) + (pbPr₁ ⋁Pullback) (DLRan F .F-hom (hom-∨₂ L C x y)) pbPr₁PathP = F≤PathPLemma ⋁β++γ≡x∨y ⋁γ≡y (subst (⋁ γ ≤_) (sym (⋁Split++ β γ)) (∨≤LCancel _ _)) (hom-∨₂ L C x y) - pbPr₂PathP : PathP (λ i → C [ DLRan F .F-ob (⋁β++γ≡x∨y i) , DLRan F .F-ob (⋁β≡x i) ]) - (pbPr₂ ⋁Pullback) (DLRan F .F-hom (hom-∨₁ L C x y)) + pbPr₂PathP : PathP (λ i → C [ DLRan F .F-ob (⋁β++γ≡x∨y i) , DLRan F .F-ob (⋁β≡x i) ]) + (pbPr₂ ⋁Pullback) (DLRan F .F-hom (hom-∨₁ L C x y)) pbPr₂PathP = F≤PathPLemma ⋁β++γ≡x∨y ⋁β≡x (subst (⋁ β ≤_) (sym (⋁Split++ β γ)) (∨≤RCancel _ _)) (hom-∨₁ L C x y) diff --git a/Cubical.Categories.Equivalence.AdjointEquivalence.html b/Cubical.Categories.Equivalence.AdjointEquivalence.html index a37445b58d..369847109c 100644 --- a/Cubical.Categories.Equivalence.AdjointEquivalence.html +++ b/Cubical.Categories.Equivalence.AdjointEquivalence.html @@ -13,81 +13,81 @@ {ℓC ℓ'C : Level} {ℓD ℓ'D : Level} where -open UnitCounit +open UnitCounit module _ (C : Category ℓC ℓ'C) (D : Category ℓD ℓ'D) where module _ - (fun : Functor C D) (inv : Functor D C) - (η : 𝟙⟨ C ⟩ ≅ᶜ inv ∘F fun) (ε : fun ∘F inv ≅ᶜ 𝟙⟨ D ⟩) - (triang : TriangleIdentities fun inv (NatIso.trans η) (NatIso.trans ε)) + (fun : Functor C D) (inv : Functor D C) + (η : 𝟙⟨ C ⟩ ≅ᶜ inv ∘F fun) (ε : fun ∘F inv ≅ᶜ 𝟙⟨ D ⟩) + (triang : TriangleIdentities fun inv (NatIso.trans η) (NatIso.trans ε)) where - open TriangleIdentities triang + open TriangleIdentities triang private module C = Category C module D = Category D - module fun = Functor fun - module inv = Functor inv + module fun = Functor fun + module inv = Functor inv - reverse-triangle : TriangleIdentities inv fun - (NatIso.trans (symNatIso ε)) - (NatIso.trans (symNatIso η)) - reverse-triangle .Δ₁ d = + reverse-triangle : TriangleIdentities inv fun + (NatIso.trans (symNatIso ε)) + (NatIso.trans (symNatIso η)) + reverse-triangle .Δ₁ d = sym (C.⋆IdR _) - ∙ cong (λ x → ( inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ - ⋆⟨ C ⟩ NatIso.trans (symNatIso η) ⟦ inv ⟅ d ⟆ ⟧) + ∙ cong (λ x → ( inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ + ⋆⟨ C ⟩ NatIso.trans (symNatIso η) ⟦ inv ⟅ d ⟆ ⟧) ⋆⟨ C ⟩ x) - (sym (Δ₂ d)) + (sym (Δ₂ d)) ∙ C.⋆Assoc _ _ _ - ∙ cong (λ x → inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ ⋆⟨ C ⟩ x) + ∙ cong (λ x → inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ ⋆⟨ C ⟩ x) (sym $ C.⋆Assoc _ _ _) - ∙ cong (λ x → inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ ⋆⟨ C ⟩ - (x ⋆⟨ C ⟩ inv ⟪ NatIso.trans ε ⟦ d ⟧ ⟫)) - (isIso.sec (NatIso.nIso η (inv ⟅ d ⟆))) - ∙ cong (λ x → inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ ⋆⟨ C ⟩ x) + ∙ cong (λ x → inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ ⋆⟨ C ⟩ + (x ⋆⟨ C ⟩ inv ⟪ NatIso.trans ε ⟦ d ⟧ ⟫)) + (isIso.sec (NatIso.nIso η (inv ⟅ d ⟆))) + ∙ cong (λ x → inv ⟪ NatIso.trans (symNatIso ε) ⟦ d ⟧ ⟫ ⋆⟨ C ⟩ x) (C.⋆IdL _) - ∙ sym (inv.F-seq _ _) - ∙ cong (λ x → inv ⟪ x ⟫) (isIso.sec (NatIso.nIso ε d)) - ∙ inv.F-id - reverse-triangle .Δ₂ c = + ∙ sym (inv.F-seq _ _) + ∙ cong (λ x → inv ⟪ x ⟫) (isIso.sec (NatIso.nIso ε d)) + ∙ inv.F-id + reverse-triangle .Δ₂ c = sym (D.⋆IdL _) ∙ cong (λ x → x - ⋆⟨ D ⟩ (NatIso.trans (symNatIso ε) ⟦ fun ⟅ c ⟆ ⟧ - ⋆⟨ D ⟩ fun ⟪ NatIso.trans (symNatIso η) ⟦ c ⟧ ⟫)) - (sym (Δ₁ c)) + ⋆⟨ D ⟩ (NatIso.trans (symNatIso ε) ⟦ fun ⟅ c ⟆ ⟧ + ⋆⟨ D ⟩ fun ⟪ NatIso.trans (symNatIso η) ⟦ c ⟧ ⟫)) + (sym (Δ₁ c)) ∙ D.⋆Assoc _ _ _ - ∙ cong (λ x → fun ⟪ NatIso.trans η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ x) + ∙ cong (λ x → fun ⟪ NatIso.trans η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ x) (sym $ D.⋆Assoc _ _ _) - ∙ cong (λ x → fun ⟪ NatIso.trans η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ - (x ⋆⟨ D ⟩ fun ⟪ NatIso.trans (symNatIso η) ⟦ c ⟧ ⟫)) - (isIso.ret (NatIso.nIso ε (fun ⟅ c ⟆))) - ∙ cong (λ x → fun ⟪ NatIso.trans η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ x) (D.⋆IdL _) - ∙ sym (fun.F-seq _ _) - ∙ cong (λ x → fun ⟪ x ⟫) (isIso.ret (NatIso.nIso η c)) - ∙ fun.F-id + ∙ cong (λ x → fun ⟪ NatIso.trans η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ + (x ⋆⟨ D ⟩ fun ⟪ NatIso.trans (symNatIso η) ⟦ c ⟧ ⟫)) + (isIso.ret (NatIso.nIso ε (fun ⟅ c ⟆))) + ∙ cong (λ x → fun ⟪ NatIso.trans η ⟦ c ⟧ ⟫ ⋆⟨ D ⟩ x) (D.⋆IdL _) + ∙ sym (fun.F-seq _ _) + ∙ cong (λ x → fun ⟪ x ⟫) (isIso.ret (NatIso.nIso η c)) + ∙ fun.F-id record AdjointEquivalence : Type (ℓ-max (ℓ-max ℓC ℓ'C) (ℓ-max ℓD ℓ'D)) where field - fun : Functor C D - inv : Functor D C - η : 𝟙⟨ C ⟩ ≅ᶜ inv ∘F fun - ε : fun ∘F inv ≅ᶜ 𝟙⟨ D ⟩ - triangleIdentities : TriangleIdentities fun inv (NatIso.trans η) (NatIso.trans ε) + fun : Functor C D + inv : Functor D C + η : 𝟙⟨ C ⟩ ≅ᶜ inv ∘F fun + ε : fun ∘F inv ≅ᶜ 𝟙⟨ D ⟩ + triangleIdentities : TriangleIdentities fun inv (NatIso.trans η) (NatIso.trans ε) module _ {C : Category ℓC ℓ'C} {D : Category ℓD ℓ'D} (adj-equiv : AdjointEquivalence C D) where open AdjointEquivalence adj-equiv - adjunction : fun ⊣ inv + adjunction : fun ⊣ inv adjunction = record { - η = NatIso.trans η - ; ε = NatIso.trans ε - ; triangleIdentities = triangleIdentities } + η = NatIso.trans η + ; ε = NatIso.trans ε + ; triangleIdentities = triangleIdentities } - reverse-adjunction : inv ⊣ fun + reverse-adjunction : inv ⊣ fun reverse-adjunction = record { - η = NatIso.trans (symNatIso ε) - ; ε = NatIso.trans (symNatIso η) - ; triangleIdentities = reverse-triangle C D fun inv η ε triangleIdentities } + η = NatIso.trans (symNatIso ε) + ; ε = NatIso.trans (symNatIso η) + ; triangleIdentities = reverse-triangle C D fun inv η ε triangleIdentities } \ No newline at end of file diff --git a/Cubical.Categories.Equivalence.Base.html b/Cubical.Categories.Equivalence.Base.html index 95a48da715..84cf9f1b18 100644 --- a/Cubical.Categories.Equivalence.Base.html +++ b/Cubical.Categories.Equivalence.Base.html @@ -11,30 +11,30 @@ open import Cubical.Categories.NaturalTransformation open Category -open Functor +open Functor private variable ℓC ℓC' ℓD ℓD' : Level record WeakInverse {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - (func : Functor C D) : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where + (func : Functor C D) : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where field - invFunc : Functor D C + invFunc : Functor D C - η : 𝟙⟨ C ⟩ ≅ᶜ invFunc ∘F func - ε : func ∘F invFunc ≅ᶜ 𝟙⟨ D ⟩ + η : 𝟙⟨ C ⟩ ≅ᶜ invFunc ∘F func + ε : func ∘F invFunc ≅ᶜ 𝟙⟨ D ⟩ -- I don't know of a good alternative representation of isEquivalence that -- avoids truncation in the general case. If the categories are univalent, then -- an adjoint equivalence might be enough. isEquivalence : {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - → (func : Functor C D) → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) + → (func : Functor C D) → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) isEquivalence func = ∥ WeakInverse func ∥₁ record _≃ᶜ_ (C : Category ℓC ℓC') (D : Category ℓD ℓD') : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where field - func : Functor C D + func : Functor C D isEquiv : isEquivalence func \ No newline at end of file diff --git a/Cubical.Categories.Equivalence.Properties.html b/Cubical.Categories.Equivalence.Properties.html index 17c7edf47d..6c09b6c63e 100644 --- a/Cubical.Categories.Equivalence.Properties.html +++ b/Cubical.Categories.Equivalence.Properties.html @@ -19,7 +19,7 @@ open import Cubical.HITs.PropositionalTruncation open Category -open Functor +open Functor open NatIso open isIso open WeakInverse @@ -31,21 +31,21 @@ -- Equivalence implies Full, Faithul, and Essentially Surjective module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} where - symWeakInverse : ∀ {F : Functor C D} → (e : WeakInverse F) → WeakInverse (e .invFunc) - symWeakInverse {F} record { invFunc = G ; η = η ; ε = ε } = record { invFunc = F ; η = symNatIso ε ; ε = symNatIso η } + symWeakInverse : ∀ {F : Functor C D} → (e : WeakInverse F) → WeakInverse (e .invFunc) + symWeakInverse {F} record { invFunc = G ; η = η ; ε = ε } = record { invFunc = F ; η = symNatIso ε ; ε = symNatIso η } - isEquiv→Faithful : ∀ {F : Functor C D} → isEquivalence F → isFaithful F + isEquiv→Faithful : ∀ {F : Functor C D} → isEquivalence F → isFaithful F isEquiv→Faithful {F} = rec (isPropΠ5 (λ _ _ _ _ _ → C .isSetHom _ _)) lifted where - lifted : WeakInverse F → isFaithful F + lifted : WeakInverse F → isFaithful F lifted record { invFunc = G ; η = η ; ε = _ } c c' f g p = f ≡⟨ sqRL η ⟩ - cIso .fst ⋆⟨ C ⟩ G ⟪ F ⟪ f ⟫ ⟫ ⋆⟨ C ⟩ c'Iso .snd .inv - ≡⟨ cong (λ v → cIso .fst ⋆⟨ C ⟩ (G ⟪ v ⟫) ⋆⟨ C ⟩ c'Iso .snd .inv) p ⟩ - cIso .fst ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⟫ ⋆⟨ C ⟩ c'Iso .snd .inv + cIso .fst ⋆⟨ C ⟩ G ⟪ F ⟪ f ⟫ ⟫ ⋆⟨ C ⟩ c'Iso .snd .inv + ≡⟨ cong (λ v → cIso .fst ⋆⟨ C ⟩ (G ⟪ v ⟫) ⋆⟨ C ⟩ c'Iso .snd .inv) p ⟩ + cIso .fst ⋆⟨ C ⟩ G ⟪ F ⟪ g ⟫ ⟫ ⋆⟨ C ⟩ c'Iso .snd .inv ≡⟨ sym (sqRL η) ⟩ g ∎ @@ -56,10 +56,10 @@ c'Iso = isIso→CatIso (η .nIso c') module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} where - isEquiv→Full : ∀ {F : Functor C D} → isEquivalence F → isFull F + isEquiv→Full : ∀ {F : Functor C D} → isEquivalence F → isFull F isEquiv→Full {F} = rec (isPropΠ3 (λ _ _ _ → isPropPropTrunc)) lifted where - lifted : WeakInverse F → isFull F + lifted : WeakInverse F → isFull F lifted eq@record { invFunc = G ; η = η ; ε = _ } @@ -74,12 +74,12 @@ cIso⁻ = symCatIso cIso c'Iso⁻ = symCatIso c'Iso - h = cIso .fst ⋆⟨ C ⟩ G ⟪ g ⟫ ⋆⟨ C ⟩ c'Iso .snd .inv + h = cIso .fst ⋆⟨ C ⟩ G ⟪ g ⟫ ⋆⟨ C ⟩ c'Iso .snd .inv -- we show that both `G ⟪ g ⟫` and `G ⟪ F ⟪ h ⟫ ⟫` -- are equal to the same thing -- namely : cIso .inv ⋆⟨ C ⟩ h ⋆⟨ C ⟩ c'Iso .mor - Gg≡ηhη : G ⟪ g ⟫ ≡ cIso .snd .inv ⋆⟨ C ⟩ h ⋆⟨ C ⟩ c'Iso .fst + Gg≡ηhη : G ⟪ g ⟫ ≡ cIso .snd .inv ⋆⟨ C ⟩ h ⋆⟨ C ⟩ c'Iso .fst Gg≡ηhη = invMoveL cAreInv move-c' ∙ sym (C .⋆Assoc _ _ _) where cAreInv : areInv _ (cIso .fst) (cIso .snd .inv) @@ -88,31 +88,31 @@ c'AreInv : areInv _ (c'Iso .fst) (c'Iso .snd .inv) c'AreInv = CatIso→areInv c'Iso - move-c' : cIso .fst ⋆⟨ C ⟩ G ⟪ g ⟫ ≡ h ⋆⟨ C ⟩ c'Iso .fst + move-c' : cIso .fst ⋆⟨ C ⟩ G ⟪ g ⟫ ≡ h ⋆⟨ C ⟩ c'Iso .fst move-c' = invMoveR (symAreInv c'AreInv) refl - GFh≡Gg : G ⟪ F ⟪ h ⟫ ⟫ ≡ G ⟪ g ⟫ - GFh≡Gg = G ⟪ F ⟪ h ⟫ ⟫ + GFh≡Gg : G ⟪ F ⟪ h ⟫ ⟫ ≡ G ⟪ g ⟫ + GFh≡Gg = G ⟪ F ⟪ h ⟫ ⟫ ≡⟨ sqLR η ⟩ cIso .snd .inv ⋆⟨ C ⟩ h ⋆⟨ C ⟩ c'Iso .fst ≡⟨ sym Gg≡ηhη ⟩ - G ⟪ g ⟫ + G ⟪ g ⟫ ∎ - isEquiv→FullyFaithful : ∀ {F : Functor C D} → isEquivalence F → isFullyFaithful F + isEquiv→FullyFaithful : ∀ {F : Functor C D} → isEquivalence F → isFullyFaithful F isEquiv→FullyFaithful {F = F} h = isFull+Faithful→isFullyFaithful {F = F} (isEquiv→Full h) (isEquiv→Faithful h) - isEquiv→Surj : ∀ {F : Functor C D} → isEquivalence F → isEssentiallySurj F + isEquiv→Surj : ∀ {F : Functor C D} → isEquivalence F → isEssentiallySurj F isEquiv→Surj = rec (isPropΠ (λ _ → isPropPropTrunc)) - (λ wkInv d → ∣ (wkInv .invFunc ⟅ d ⟆) , isIso→CatIso ((wkInv .ε .nIso) d) ∣₁) + (λ wkInv d → ∣ (wkInv .invFunc ⟅ d ⟆) , isIso→CatIso ((wkInv .ε .nIso) d) ∣₁) -- A fully-faithful functor that induces equivalence on objects is an equivalence module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - {F : Functor C D} where + {F : Functor C D} where - isFullyFaithful+isEquivF-ob→isEquiv : isFullyFaithful F → isEquivMap (F .F-ob) → isEquivalence F + isFullyFaithful+isEquivF-ob→isEquiv : isFullyFaithful F → isEquivMap (F .F-ob) → isEquivalence F isFullyFaithful+isEquivF-ob→isEquiv fullfaith isequiv = ∣ w ∣₁ where open Iso @@ -124,19 +124,19 @@ MorD : D .ob × D .ob → Type _ MorD (x , y) = D [ x , y ] - F-Mor : ((x , y) : C .ob × C .ob) → C [ x , y ] → D [ F .F-ob x , F .F-ob y ] - F-Mor _ = F .F-hom + F-Mor : ((x , y) : C .ob × C .ob) → C [ x , y ] → D [ F .F-ob x , F .F-ob y ] + F-Mor _ = F .F-hom equiv-ob² : C .ob × C .ob ≃ D .ob × D .ob - equiv-ob² = ≃-× (_ , isequiv) (_ , isequiv) + equiv-ob² = ≃-× (_ , isequiv) (_ , isequiv) iso-ob = equivToIso (_ , isequiv) iso-hom = equivOver→IsoOver {P = MorC} {Q = MorD} equiv-ob² F-Mor (λ (x , y) → fullfaith x y) - w-inv : Functor D C - w-inv .F-ob = iso-ob .inv - w-inv .F-hom = iso-hom .inv _ - w-inv .F-id {x = x} = isFullyFaithful→Faithful {F = F} fullfaith _ _ _ _ (p ∙ sym (F .F-id)) + w-inv : Functor D C + w-inv .F-ob = iso-ob .inv + w-inv .F-hom = iso-hom .inv _ + w-inv .F-id {x = x} = isFullyFaithful→Faithful {F = F} fullfaith _ _ _ _ (p ∙ sym (F .F-id)) where p : _ p i = @@ -146,7 +146,7 @@ { (i = i0) → iso-hom .rightInv _ (D .id {x = x}) (~ j) ; (i = i1) → D .id {x = iso-ob .rightInv x (~ j)} }) (D .id {x = x}) - w-inv .F-seq {x = x} {z = z} f g = isFullyFaithful→Faithful {F = F} fullfaith _ _ _ _ (p ∙ sym (F .F-seq _ _)) + w-inv .F-seq {x = x} {z = z} f g = isFullyFaithful→Faithful {F = F} fullfaith _ _ _ _ (p ∙ sym (F .F-seq _ _)) where p : _ p i = @@ -157,11 +157,11 @@ ; (i = i1) → iso-hom .rightInv _ f (~ j) ⋆⟨ D ⟩ iso-hom .rightInv _ g (~ j) }) (f ⋆⟨ D ⟩ g) - w-η-path : 𝟙⟨ C ⟩ ≡ w-inv ∘F F - w-η-path = Functor≡ (λ x → sym (retIsEq isequiv x)) (λ {x} {y} f → (λ i → iso-hom .leftInv (x , y) f (~ i))) + w-η-path : 𝟙⟨ C ⟩ ≡ w-inv ∘F F + w-η-path = Functor≡ (λ x → sym (retIsEq isequiv x)) (λ {x} {y} f → (λ i → iso-hom .leftInv (x , y) f (~ i))) - w-ε-path : F ∘F w-inv ≡ 𝟙⟨ D ⟩ - w-ε-path = Functor≡ (λ x → secIsEq isequiv x) (λ {x} {y} f i → iso-hom .rightInv (x , y) f i) + w-ε-path : F ∘F w-inv ≡ 𝟙⟨ D ⟩ + w-ε-path = Functor≡ (λ x → secIsEq isequiv x) (λ {x} {y} f i → iso-hom .rightInv (x , y) f i) w : WeakInverse F w .invFunc = w-inv @@ -171,7 +171,7 @@ -- equivalence on full subcategories defined by propositions -module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) (invF : WeakInverse F) where +module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) (invF : WeakInverse F) where open NatTrans open _≃ᶜ_ @@ -183,11 +183,11 @@ ΣPropCatEquiv : {P : ℙ (ob C)} {Q : ℙ (ob D)} - → (presF : ∀ c → c ∈ P → F .F-ob c ∈ Q) - → (∀ d → d ∈ Q → F⁻¹ .F-ob d ∈ P) - → WeakInverse (ΣPropCatFunc {P = P} {Q = Q} F presF) + → (presF : ∀ c → c ∈ P → F .F-ob c ∈ Q) + → (∀ d → d ∈ Q → F⁻¹ .F-ob d ∈ P) + → WeakInverse (ΣPropCatFunc {P = P} {Q = Q} F presF) - invFunc (ΣPropCatEquiv {P} {Q} _ presF⁻¹) = ΣPropCatFunc {P = Q} {Q = P} F⁻¹ presF⁻¹ + invFunc (ΣPropCatEquiv {P} {Q} _ presF⁻¹) = ΣPropCatFunc {P = Q} {Q = P} F⁻¹ presF⁻¹ N-ob (trans (η (ΣPropCatEquiv _ _))) (x , _) = ηᴱ .trans .N-ob x N-hom (trans (η (ΣPropCatEquiv _ _))) f = ηᴱ .trans .N-hom f diff --git a/Cubical.Categories.Equivalence.WeakEquivalence.html b/Cubical.Categories.Equivalence.WeakEquivalence.html index b730639efd..ca4bb65128 100644 --- a/Cubical.Categories.Equivalence.WeakEquivalence.html +++ b/Cubical.Categories.Equivalence.WeakEquivalence.html @@ -21,26 +21,26 @@ ℓC ℓC' ℓD ℓD' : Level C : Category ℓC ℓC' D : Category ℓD ℓD' - F : Functor C D + F : Functor C D -open Functor +open Functor -- Weak equivalences of categories, -- namely fully-faithful and essentially surjective functors record isWeakEquivalence {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - (func : Functor C D) : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where + (func : Functor C D) : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where field - fullfaith : isFullyFaithful func - esssurj : isEssentiallySurj func + fullfaith : isFullyFaithful func + esssurj : isEssentiallySurj func record WeakEquivalence (C : Category ℓC ℓC') (D : Category ℓD ℓD') : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where field - func : Functor C D + func : Functor C D isWeakEquiv : isWeakEquivalence func open isWeakEquivalence @@ -64,12 +64,12 @@ open isUnivalent - isEquivF-ob : {F : Functor C D} → isWeakEquivalence F → isEquivMap (F .F-ob) + isEquivF-ob : {F : Functor C D} → isWeakEquivalence F → isEquivMap (F .F-ob) isEquivF-ob {F = F} is-w-equiv = isEmbedding×isSurjection→isEquiv (isFullyFaithful→isEmbd-ob isUnivC isUnivD {F = F} (is-w-equiv .fullfaith) , isSurj→isSurj-ob isUnivD {F = F} (is-w-equiv .esssurj)) - isWeakEquiv→isEquiv : {F : Functor C D} → isWeakEquivalence F → isEquivalence F + isWeakEquiv→isEquiv : {F : Functor C D} → isWeakEquivalence F → isEquivalence F isWeakEquiv→isEquiv is-w-equiv = isFullyFaithful+isEquivF-ob→isEquiv (is-w-equiv .fullfaith) (isEquivF-ob is-w-equiv) \ No newline at end of file diff --git a/Cubical.Categories.Everything.html b/Cubical.Categories.Everything.html index 6ff1e40d31..3b57f877e2 100644 --- a/Cubical.Categories.Everything.html +++ b/Cubical.Categories.Everything.html @@ -56,31 +56,32 @@ import Cubical.Categories.Instances.FunctorAlgebras import Cubical.Categories.Instances.Functors import Cubical.Categories.Instances.Functors.Endo -import Cubical.Categories.Instances.Lattice -import Cubical.Categories.Instances.Lattices -import Cubical.Categories.Instances.Poset -import Cubical.Categories.Instances.RingAlgebras -import Cubical.Categories.Instances.Rings -import Cubical.Categories.Instances.Semilattice -import Cubical.Categories.Instances.Sets -import Cubical.Categories.Instances.Terminal -import Cubical.Categories.Instances.TypePrecategory -import Cubical.Categories.Instances.ZFunctors -import Cubical.Categories.Isomorphism -import Cubical.Categories.Limits -import Cubical.Categories.Limits.RightKan -import Cubical.Categories.Monad.Base -import Cubical.Categories.Monoidal -import Cubical.Categories.Monoidal.Strict.Monoid -import Cubical.Categories.Morphism -import Cubical.Categories.NaturalTransformation -import Cubical.Categories.Presheaf -import Cubical.Categories.Presheaf.NonPresheaf.Cofree -import Cubical.Categories.Presheaf.NonPresheaf.Forget -import Cubical.Categories.Presheaf.NonPresheaf.Free -import Cubical.Categories.Profunctor -import Cubical.Categories.RezkCompletion -import Cubical.Categories.TypesOfCategories.TypeCategory -import Cubical.Categories.UnderlyingGraph -import Cubical.Categories.Yoneda +import Cubical.Categories.Instances.Groups +import Cubical.Categories.Instances.Lattice +import Cubical.Categories.Instances.Lattices +import Cubical.Categories.Instances.Poset +import Cubical.Categories.Instances.RingAlgebras +import Cubical.Categories.Instances.Rings +import Cubical.Categories.Instances.Semilattice +import Cubical.Categories.Instances.Sets +import Cubical.Categories.Instances.Terminal +import Cubical.Categories.Instances.TypePrecategory +import Cubical.Categories.Instances.ZFunctors +import Cubical.Categories.Isomorphism +import Cubical.Categories.Limits +import Cubical.Categories.Limits.RightKan +import Cubical.Categories.Monad.Base +import Cubical.Categories.Monoidal +import Cubical.Categories.Monoidal.Strict.Monoid +import Cubical.Categories.Morphism +import Cubical.Categories.NaturalTransformation +import Cubical.Categories.Presheaf +import Cubical.Categories.Presheaf.NonPresheaf.Cofree +import Cubical.Categories.Presheaf.NonPresheaf.Forget +import Cubical.Categories.Presheaf.NonPresheaf.Free +import Cubical.Categories.Profunctor +import Cubical.Categories.RezkCompletion +import Cubical.Categories.TypesOfCategories.TypeCategory +import Cubical.Categories.UnderlyingGraph +import Cubical.Categories.Yoneda \ No newline at end of file diff --git a/Cubical.Categories.Functor.Base.html b/Cubical.Categories.Functor.Base.html index fb0f47f889..9606cd6fd2 100644 --- a/Cubical.Categories.Functor.Base.html +++ b/Cubical.Categories.Functor.Base.html @@ -7,155 +7,172 @@ open import Cubical.Foundations.Equiv open import Cubical.Foundations.HLevels open import Cubical.Foundations.Powerset - -open import Cubical.Data.Sigma - -open import Cubical.Categories.Category - -private - variable - ℓC ℓC' ℓD ℓD' : Level - -record Functor (C : Category ℓC ℓC') (D : Category ℓD ℓD') : - Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where - no-eta-equality - - open Category - - field - F-ob : C .ob → D .ob - F-hom : {x y : C .ob} → C [ x , y ] → D [ F-ob x , F-ob y ] - F-id : {x : C .ob} → F-hom (C .id) ≡ D .id {x = F-ob x} - F-seq : {x y z : C .ob} (f : C [ x , y ]) (g : C [ y , z ]) - → F-hom (f ⋆⟨ C ⟩ g) ≡ (F-hom f) ⋆⟨ D ⟩ (F-hom g) - - isFull = (x y : _) (F[f] : D [ F-ob x , F-ob y ]) → ∃[ f ∈ C [ x , y ] ] F-hom f ≡ F[f] - isFaithful = (x y : _) (f g : C [ x , y ]) → F-hom f ≡ F-hom g → f ≡ g - isFullyFaithful = (x y : _) → isEquiv (F-hom {x = x} {y = y}) - isEssentiallySurj = (d : D .ob) → ∃[ c ∈ C .ob ] CatIso D (F-ob c) d - - -- preservation of commuting squares and triangles - F-square : {x y u v : C .ob} - {f : C [ x , y ]} {g : C [ x , u ]} - {h : C [ u , v ]} {k : C [ y , v ]} - → f ⋆⟨ C ⟩ k ≡ g ⋆⟨ C ⟩ h - → (F-hom f) ⋆⟨ D ⟩ (F-hom k) ≡ (F-hom g) ⋆⟨ D ⟩ (F-hom h) - F-square Csquare = sym (F-seq _ _) ∙∙ cong F-hom Csquare ∙∙ F-seq _ _ - - F-triangle : {x y z : C .ob} - {f : C [ x , y ]} {g : C [ y , z ]} {h : C [ x , z ]} - → f ⋆⟨ C ⟩ g ≡ h - → (F-hom f) ⋆⟨ D ⟩ (F-hom g) ≡ (F-hom h) - F-triangle Ctriangle = sym (F-seq _ _) ∙ cong F-hom Ctriangle - -private - variable - ℓ ℓ' : Level - B C D E : Category ℓ ℓ' - -open Category -open Functor - -Functor≡ : {F G : Functor C D} - → (h : ∀ (c : ob C) → F-ob F c ≡ F-ob G c) - → (∀ {c c' : ob C} (f : C [ c , c' ]) → PathP (λ i → D [ h c i , h c' i ]) (F-hom F f) (F-hom G f)) - → F ≡ G -F-ob (Functor≡ hOb hHom i) c = hOb c i -F-hom (Functor≡ hOb hHom i) f = hHom f i -F-id (Functor≡ {C = C} {D = D} {F = F} {G = G} hOb hHom i) = - isProp→PathP (λ j → isSetHom D (hHom (C .id) j) (D .id)) (F-id F) (F-id G) i -F-seq (Functor≡ {C = C} {D = D} {F = F} {G = G} hOb hHom i) f g = - isProp→PathP (λ j → isSetHom D (hHom (f ⋆⟨ C ⟩ g) j) ((hHom f j) ⋆⟨ D ⟩ (hHom g j))) (F-seq F f g) (F-seq G f g) i - -FunctorSquare : - {F₀₀ F₀₁ F₁₀ F₁₁ : Functor C D} - (F₀₋ : F₀₀ ≡ F₀₁) (F₁₋ : F₁₀ ≡ F₁₁) - (F₋₀ : F₀₀ ≡ F₁₀) (F₋₁ : F₀₁ ≡ F₁₁) - → Square (cong F-ob F₀₋) (cong F-ob F₁₋) (cong F-ob F₋₀) (cong F-ob F₋₁) - → Square F₀₋ F₁₋ F₋₀ F₋₁ -FunctorSquare {C = C} {D = D} F₀₋ F₁₋ F₋₀ F₋₁ r = sqr - where - sqr : _ - sqr i j .F-ob = r i j - sqr i j .F-hom {x = x} {y = y} f = - isSet→SquareP (λ i j → D .isSetHom {x = sqr i j .F-ob x} {y = sqr i j .F-ob y}) - (λ i → F₀₋ i .F-hom f) (λ i → F₁₋ i .F-hom f) (λ i → F₋₀ i .F-hom f) (λ i → F₋₁ i .F-hom f) i j - sqr i j .F-id {x = x} = - isSet→SquareP (λ i j → isProp→isSet (D .isSetHom (sqr i j .F-hom (C .id)) (D .id))) - (λ i → F₀₋ i .F-id) (λ i → F₁₋ i .F-id) (λ i → F₋₀ i .F-id) (λ i → F₋₁ i .F-id) i j - sqr i j .F-seq f g = - isSet→SquareP (λ i j → - isProp→isSet (D .isSetHom (sqr i j .F-hom (f ⋆⟨ C ⟩ g)) ((sqr i j .F-hom f) ⋆⟨ D ⟩ (sqr i j .F-hom g)))) - (λ i → F₀₋ i .F-seq f g) (λ i → F₁₋ i .F-seq f g) (λ i → F₋₀ i .F-seq f g) (λ i → F₋₁ i .F-seq f g) i j - -FunctorPath≡ : {F G : Functor C D}{p q : F ≡ G} → cong F-ob p ≡ cong F-ob q → p ≡ q -FunctorPath≡ {p = p} {q = q} = FunctorSquare p q refl refl - - --- Helpful notation - --- action on objects -infix 30 _⟅_⟆ -_⟅_⟆ : (F : Functor C D) - → C .ob - → D .ob -_⟅_⟆ = F-ob - --- action on morphisms -infix 30 _⟪_⟫ -- same infix level as on objects since these will never be used in the same context -_⟪_⟫ : (F : Functor C D) - → ∀ {x y} - → C [ x , y ] - → D [(F ⟅ x ⟆) , (F ⟅ y ⟆)] -_⟪_⟫ = F-hom - - --- Functor constructions - -𝟙⟨_⟩ : ∀ (C : Category ℓ ℓ') → Functor C C -𝟙⟨ C ⟩ .F-ob x = x -𝟙⟨ C ⟩ .F-hom f = f -𝟙⟨ C ⟩ .F-id = refl -𝟙⟨ C ⟩ .F-seq _ _ = refl - -Id : {C : Category ℓ ℓ'} → Functor C C -Id = 𝟙⟨ _ ⟩ - -forgetΣPropCat : (C : Category ℓ ℓ') (prop : ℙ (C .ob)) → Functor (ΣPropCat C prop) C -forgetΣPropCat _ _ .F-ob x = x .fst -forgetΣPropCat _ _ .F-hom f = f -forgetΣPropCat _ _ .F-id = refl -forgetΣPropCat _ _ .F-seq _ _ = refl - --- functor composition -funcComp : ∀ (G : Functor D E) (F : Functor C D) → Functor C E -(funcComp G F) .F-ob c = G ⟅ F ⟅ c ⟆ ⟆ -(funcComp G F) .F-hom f = G ⟪ F ⟪ f ⟫ ⟫ -(funcComp G F) .F-id = cong (G ⟪_⟫) (F .F-id) ∙ G .F-id -(funcComp G F) .F-seq f g = cong (G ⟪_⟫) (F .F-seq _ _) ∙ G .F-seq _ _ - -infixr 30 _∘F_ -_∘F_ : Functor D E → Functor C D → Functor C E -_∘F_ = funcComp - --- hacky lemma to stop Agda from computing too much -funcCompOb≡ : ∀ (G : Functor D E) (F : Functor C D) (c : ob C) - → funcComp G F .F-ob c ≡ G .F-ob (F .F-ob c) -funcCompOb≡ G F c = refl - -_^opF : Functor C D → Functor (C ^op) (D ^op) -(F ^opF) .F-ob = F .F-ob -(F ^opF) .F-hom = F .F-hom -(F ^opF) .F-id = F .F-id -(F ^opF) .F-seq f g = F .F-seq g f - - --- Functoriality on full subcategories defined by propositions -ΣPropCatFunc : {P : ℙ (ob C)} {Q : ℙ (ob D)} (F : Functor C D) - → (∀ c → c ∈ P → F .F-ob c ∈ Q) - → Functor (ΣPropCat C P) (ΣPropCat D Q) -F-ob (ΣPropCatFunc F FPres) (c , c∈P) = F .F-ob c , FPres c c∈P -F-hom (ΣPropCatFunc F FPres) = F .F-hom -F-id (ΣPropCatFunc F FPres) = F .F-id -F-seq (ΣPropCatFunc F FPres) = F .F-seq +open import Cubical.Foundations.Isomorphism + +open import Cubical.Data.Sigma + +open import Cubical.Categories.Category + +private + variable + ℓC ℓC' ℓD ℓD' : Level + +record Functor (C : Category ℓC ℓC') (D : Category ℓD ℓD') : + Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where + no-eta-equality + + open Category + + field + F-ob : C .ob → D .ob + F-hom : {x y : C .ob} → C [ x , y ] → D [ F-ob x , F-ob y ] + F-id : {x : C .ob} → F-hom (C .id) ≡ D .id {x = F-ob x} + F-seq : {x y z : C .ob} (f : C [ x , y ]) (g : C [ y , z ]) + → F-hom (f ⋆⟨ C ⟩ g) ≡ (F-hom f) ⋆⟨ D ⟩ (F-hom g) + + isFull = (x y : _) (F[f] : D [ F-ob x , F-ob y ]) → ∃[ f ∈ C [ x , y ] ] F-hom f ≡ F[f] + isFaithful = (x y : _) (f g : C [ x , y ]) → F-hom f ≡ F-hom g → f ≡ g + isFullyFaithful = (x y : _) → isEquiv (F-hom {x = x} {y = y}) + isEssentiallySurj = (d : D .ob) → ∃[ c ∈ C .ob ] CatIso D (F-ob c) d + + -- preservation of commuting squares and triangles + F-square : {x y u v : C .ob} + {f : C [ x , y ]} {g : C [ x , u ]} + {h : C [ u , v ]} {k : C [ y , v ]} + → f ⋆⟨ C ⟩ k ≡ g ⋆⟨ C ⟩ h + → (F-hom f) ⋆⟨ D ⟩ (F-hom k) ≡ (F-hom g) ⋆⟨ D ⟩ (F-hom h) + F-square Csquare = sym (F-seq _ _) ∙∙ cong F-hom Csquare ∙∙ F-seq _ _ + + F-triangle : {x y z : C .ob} + {f : C [ x , y ]} {g : C [ y , z ]} {h : C [ x , z ]} + → f ⋆⟨ C ⟩ g ≡ h + → (F-hom f) ⋆⟨ D ⟩ (F-hom g) ≡ (F-hom h) + F-triangle Ctriangle = sym (F-seq _ _) ∙ cong F-hom Ctriangle + +private + variable + ℓ ℓ' : Level + B C D E : Category ℓ ℓ' + +open Category +open Functor + +Functor≡ : {F G : Functor C D} + → (h : ∀ (c : ob C) → F-ob F c ≡ F-ob G c) + → (∀ {c c' : ob C} (f : C [ c , c' ]) → PathP (λ i → D [ h c i , h c' i ]) (F-hom F f) (F-hom G f)) + → F ≡ G +F-ob (Functor≡ hOb hHom i) c = hOb c i +F-hom (Functor≡ hOb hHom i) f = hHom f i +F-id (Functor≡ {C = C} {D = D} {F = F} {G = G} hOb hHom i) = + isProp→PathP (λ j → isSetHom D (hHom (C .id) j) (D .id)) (F-id F) (F-id G) i +F-seq (Functor≡ {C = C} {D = D} {F = F} {G = G} hOb hHom i) f g = + isProp→PathP (λ j → isSetHom D (hHom (f ⋆⟨ C ⟩ g) j) ((hHom f j) ⋆⟨ D ⟩ (hHom g j))) (F-seq F f g) (F-seq G f g) i + +FunctorSquare : + {F₀₀ F₀₁ F₁₀ F₁₁ : Functor C D} + (F₀₋ : F₀₀ ≡ F₀₁) (F₁₋ : F₁₀ ≡ F₁₁) + (F₋₀ : F₀₀ ≡ F₁₀) (F₋₁ : F₀₁ ≡ F₁₁) + → Square (cong F-ob F₀₋) (cong F-ob F₁₋) (cong F-ob F₋₀) (cong F-ob F₋₁) + → Square F₀₋ F₁₋ F₋₀ F₋₁ +FunctorSquare {C = C} {D = D} F₀₋ F₁₋ F₋₀ F₋₁ r = sqr + where + sqr : _ + sqr i j .F-ob = r i j + sqr i j .F-hom {x = x} {y = y} f = + isSet→SquareP (λ i j → D .isSetHom {x = sqr i j .F-ob x} {y = sqr i j .F-ob y}) + (λ i → F₀₋ i .F-hom f) (λ i → F₁₋ i .F-hom f) (λ i → F₋₀ i .F-hom f) (λ i → F₋₁ i .F-hom f) i j + sqr i j .F-id {x = x} = + isSet→SquareP (λ i j → isProp→isSet (D .isSetHom (sqr i j .F-hom (C .id)) (D .id))) + (λ i → F₀₋ i .F-id) (λ i → F₁₋ i .F-id) (λ i → F₋₀ i .F-id) (λ i → F₋₁ i .F-id) i j + sqr i j .F-seq f g = + isSet→SquareP (λ i j → + isProp→isSet (D .isSetHom (sqr i j .F-hom (f ⋆⟨ C ⟩ g)) ((sqr i j .F-hom f) ⋆⟨ D ⟩ (sqr i j .F-hom g)))) + (λ i → F₀₋ i .F-seq f g) (λ i → F₁₋ i .F-seq f g) (λ i → F₋₀ i .F-seq f g) (λ i → F₋₁ i .F-seq f g) i j + +FunctorPath≡ : {F G : Functor C D}{p q : F ≡ G} → cong F-ob p ≡ cong F-ob q → p ≡ q +FunctorPath≡ {p = p} {q = q} = FunctorSquare p q refl refl + + +-- Helpful notation + +-- action on objects +infix 30 _⟅_⟆ +_⟅_⟆ : (F : Functor C D) + → C .ob + → D .ob +_⟅_⟆ = F-ob + +-- action on morphisms +infix 30 _⟪_⟫ -- same infix level as on objects since these will never be used in the same context +_⟪_⟫ : (F : Functor C D) + → ∀ {x y} + → C [ x , y ] + → D [(F ⟅ x ⟆) , (F ⟅ y ⟆)] +_⟪_⟫ = F-hom + + +-- Functor constructions + +𝟙⟨_⟩ : ∀ (C : Category ℓ ℓ') → Functor C C +𝟙⟨ C ⟩ .F-ob x = x +𝟙⟨ C ⟩ .F-hom f = f +𝟙⟨ C ⟩ .F-id = refl +𝟙⟨ C ⟩ .F-seq _ _ = refl + +Id : {C : Category ℓ ℓ'} → Functor C C +Id = 𝟙⟨ _ ⟩ + +forgetΣPropCat : (C : Category ℓ ℓ') (prop : ℙ (C .ob)) → Functor (ΣPropCat C prop) C +forgetΣPropCat _ _ .F-ob x = x .fst +forgetΣPropCat _ _ .F-hom f = f +forgetΣPropCat _ _ .F-id = refl +forgetΣPropCat _ _ .F-seq _ _ = refl + +-- functor composition +funcComp : ∀ (G : Functor D E) (F : Functor C D) → Functor C E +(funcComp G F) .F-ob c = G ⟅ F ⟅ c ⟆ ⟆ +(funcComp G F) .F-hom f = G ⟪ F ⟪ f ⟫ ⟫ +(funcComp G F) .F-id = cong (G ⟪_⟫) (F .F-id) ∙ G .F-id +(funcComp G F) .F-seq f g = cong (G ⟪_⟫) (F .F-seq _ _) ∙ G .F-seq _ _ + +infixr 30 _∘F_ +_∘F_ : Functor D E → Functor C D → Functor C E +_∘F_ = funcComp + +-- hacky lemma to stop Agda from computing too much +funcCompOb≡ : ∀ (G : Functor D E) (F : Functor C D) (c : ob C) + → funcComp G F .F-ob c ≡ G .F-ob (F .F-ob c) +funcCompOb≡ G F c = refl + + +_^opF : Functor C D → Functor (C ^op) (D ^op) +(F ^opF) .F-ob = F .F-ob +(F ^opF) .F-hom = F .F-hom +(F ^opF) .F-id = F .F-id +(F ^opF) .F-seq f g = F .F-seq g f + +open Iso +Iso^opF : Iso (Functor C D) (Functor (C ^op) (D ^op)) +fun Iso^opF = _^opF +inv Iso^opF = _^opF +F-ob (rightInv Iso^opF b i) = F-ob b +F-hom (rightInv Iso^opF b i) = F-hom b +F-id (rightInv Iso^opF b i) = F-id b +F-seq (rightInv Iso^opF b i) = F-seq b +F-ob (leftInv Iso^opF a i) = F-ob a +F-hom (leftInv Iso^opF a i) = F-hom a +F-id (leftInv Iso^opF a i) = F-id a +F-seq (leftInv Iso^opF a i) = F-seq a + +^opFEquiv : Functor C D ≃ Functor (C ^op) (D ^op) +^opFEquiv = isoToEquiv Iso^opF + +-- Functoriality on full subcategories defined by propositions +ΣPropCatFunc : {P : ℙ (ob C)} {Q : ℙ (ob D)} (F : Functor C D) + → (∀ c → c ∈ P → F .F-ob c ∈ Q) + → Functor (ΣPropCat C P) (ΣPropCat D Q) +F-ob (ΣPropCatFunc F FPres) (c , c∈P) = F .F-ob c , FPres c c∈P +F-hom (ΣPropCatFunc F FPres) = F .F-hom +F-id (ΣPropCatFunc F FPres) = F .F-id +F-seq (ΣPropCatFunc F FPres) = F .F-seq \ No newline at end of file diff --git a/Cubical.Categories.Functor.Compose.html b/Cubical.Categories.Functor.Compose.html index 157c267302..1c29e9118f 100644 --- a/Cubical.Categories.Functor.Compose.html +++ b/Cubical.Categories.Functor.Compose.html @@ -13,34 +13,34 @@ module _ {ℓC ℓC' ℓD ℓD' ℓE ℓE'} {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (E : Category ℓE ℓE') - (F : Functor C D) + (F : Functor C D) where - open Functor + open Functor open NatTrans - precomposeF : Functor (FUNCTOR D E) (FUNCTOR C E) - precomposeF .F-ob G = funcComp G F - precomposeF .F-hom α .N-ob c = α .N-ob (F .F-ob c) - precomposeF .F-hom α .N-hom f = α .N-hom (F .F-hom f) - precomposeF .F-id = refl - precomposeF .F-seq f g = refl + precomposeF : Functor (FUNCTOR D E) (FUNCTOR C E) + precomposeF .F-ob G = funcComp G F + precomposeF .F-hom α .N-ob c = α .N-ob (F .F-ob c) + precomposeF .F-hom α .N-hom f = α .N-hom (F .F-hom f) + precomposeF .F-id = refl + precomposeF .F-seq f g = refl module _ {ℓC ℓC' ℓD ℓD' ℓE ℓE'} (C : Category ℓC ℓC') {D : Category ℓD ℓD'} {E : Category ℓE ℓE'} - (G : Functor D E) + (G : Functor D E) where - open Functor + open Functor open NatTrans - postcomposeF : Functor (FUNCTOR C D) (FUNCTOR C E) - postcomposeF .F-ob F = funcComp G F - postcomposeF .F-hom α .N-ob c = G. F-hom (α .N-ob c) - postcomposeF .F-hom {F₀} {F₁} α .N-hom {x} {y} f = - sym (G .F-seq (F₀ ⟪ f ⟫) (α ⟦ y ⟧)) - ∙∙ cong (G ⟪_⟫) (α .N-hom f) - ∙∙ G .F-seq (α ⟦ x ⟧) (F₁ ⟪ f ⟫) - postcomposeF .F-id = makeNatTransPath (funExt λ _ → G .F-id) - postcomposeF .F-seq f g = makeNatTransPath (funExt λ _ → G .F-seq _ _) + postcomposeF : Functor (FUNCTOR C D) (FUNCTOR C E) + postcomposeF .F-ob F = funcComp G F + postcomposeF .F-hom α .N-ob c = G. F-hom (α .N-ob c) + postcomposeF .F-hom {F₀} {F₁} α .N-hom {x} {y} f = + sym (G .F-seq (F₀ ⟪ f ⟫) (α ⟦ y ⟧)) + ∙∙ cong (G ⟪_⟫) (α .N-hom f) + ∙∙ G .F-seq (α ⟦ x ⟧) (F₁ ⟪ f ⟫) + postcomposeF .F-id = makeNatTransPath (funExt λ _ → G .F-id) + postcomposeF .F-seq f g = makeNatTransPath (funExt λ _ → G .F-seq _ _) \ No newline at end of file diff --git a/Cubical.Categories.Functor.ComposeProperty.html b/Cubical.Categories.Functor.ComposeProperty.html index 694639df7f..8f07798047 100644 --- a/Cubical.Categories.Functor.ComposeProperty.html +++ b/Cubical.Categories.Functor.ComposeProperty.html @@ -24,41 +24,41 @@ module _ {ℓC ℓC' ℓD ℓD' ℓE ℓE'} {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (E : Category ℓE ℓE') - (F : Functor C D) + (F : Functor C D) where open Category - open Functor + open Functor open NatTrans open isIso -- If F is essential surjective, (- ∘ F) is faithful. - isEssSurj→isFaithfulPrecomp : isEssentiallySurj F → isFaithful (precomposeF E F) + isEssSurj→isFaithfulPrecomp : isEssentiallySurj F → isFaithful (precomposeF E F) isEssSurj→isFaithfulPrecomp surj A B α β p = makeNatTransPath (λ i x → Prop.rec (E .isSetHom _ _) (λ (c , f) → ⋆InvLMove (F-Iso {F = A} f) (α .N-hom (f .fst)) - ∙ (λ i → F-Iso {F = A} f .snd .inv ⋆⟨ E ⟩ (p i .N-ob c ⋆⟨ E ⟩ B .F-hom (f .fst))) + ∙ (λ i → F-Iso {F = A} f .snd .inv ⋆⟨ E ⟩ (p i .N-ob c ⋆⟨ E ⟩ B .F-hom (f .fst))) ∙ sym (⋆InvLMove (F-Iso {F = A} f) (β .N-hom (f .fst)))) (surj x) i) -- If F is essential surjective and full, (- ∘ F) is full. - isEssSurj+Full→isFullPrecomp : isEssentiallySurj F → isFull F → isFull (precomposeF E F) + isEssSurj+Full→isFullPrecomp : isEssentiallySurj F → isFull F → isFull (precomposeF E F) isEssSurj+Full→isFullPrecomp surj full A B α = ∣ ext , ext≡ ∣₁ where Mor : (d : D .ob) → Type _ Mor d = - Σ[ g ∈ E [ A .F-ob d , B .F-ob d ] ] - ((c : C .ob)(f : CatIso D (F .F-ob c) d) - → α .N-ob c ≡ A .F-hom (f .fst) ⋆⟨ E ⟩ g ⋆⟨ E ⟩ B .F-hom (f .snd .inv)) + Σ[ g ∈ E [ A .F-ob d , B .F-ob d ] ] + ((c : C .ob)(f : CatIso D (F .F-ob c) d) + → α .N-ob c ≡ A .F-hom (f .fst) ⋆⟨ E ⟩ g ⋆⟨ E ⟩ B .F-hom (f .snd .inv)) isPropMor : (d : D .ob) → isProp (Mor d) - isPropMor d x y = Σ≡Prop (λ _ → isPropΠ2 (λ _ _ → E .isSetHom _ _)) path + isPropMor d x y = Σ≡Prop (λ _ → isPropΠ2 (λ _ _ → E .isSetHom _ _)) path where path : x .fst ≡ y .fst path = Prop.rec (E .isSetHom _ _) @@ -73,69 +73,69 @@ inhab : Mor d inhab = Prop.rec (isPropMor d) (λ (a , h) → - A .F-hom (h .snd .inv) ⋆⟨ E ⟩ α .N-ob a ⋆⟨ E ⟩ B .F-hom (h .fst) , + A .F-hom (h .snd .inv) ⋆⟨ E ⟩ α .N-ob a ⋆⟨ E ⟩ B .F-hom (h .fst) , λ c f → Prop.rec (E .isSetHom _ _) (λ (k , p) → let isom-path = subst-filler (isIso D) (sym p) (⋆Iso f (invIso h) .snd) in ⋆InvRMove (F-Iso {F = B} (_ , isom-path i1)) (sym (α .N-hom k)) - ∙ (λ i → A .F-hom (p i) ⋆⟨ E ⟩ α .N-ob a ⋆⟨ E ⟩ F-Iso {F = B} (p i , isom-path (~ i)) .snd .inv) - ∙ (λ i → A .F-seq (f .fst) (h .snd .inv) i ⋆⟨ E ⟩ α .N-ob a ⋆⟨ E ⟩ F-Iso-Pres⋆ {F = B} h (invIso f) i .fst) + ∙ (λ i → A .F-hom (p i) ⋆⟨ E ⟩ α .N-ob a ⋆⟨ E ⟩ F-Iso {F = B} (p i , isom-path (~ i)) .snd .inv) + ∙ (λ i → A .F-seq (f .fst) (h .snd .inv) i ⋆⟨ E ⟩ α .N-ob a ⋆⟨ E ⟩ F-Iso-Pres⋆ {F = B} h (invIso f) i .fst) ∙ sym (E .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (E .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ E ⟩ _) (E .⋆Assoc _ _ _)) (full _ _ (f .fst ⋆⟨ D ⟩ h .snd .inv))) (surj d) - mor-eq : (d : D .ob)(c : C .ob)(f : CatIso D (F .F-ob c) d) - → isContrMor d .fst .fst ≡ A .F-hom (f .snd .inv) ⋆⟨ E ⟩ α .N-ob c ⋆⟨ E ⟩ B .F-hom (f .fst) + mor-eq : (d : D .ob)(c : C .ob)(f : CatIso D (F .F-ob c) d) + → isContrMor d .fst .fst ≡ A .F-hom (f .snd .inv) ⋆⟨ E ⟩ α .N-ob c ⋆⟨ E ⟩ B .F-hom (f .fst) mor-eq d c f = ⋆InvLMove (F-Iso {F = A} f) (⋆InvRMove (invIso (F-Iso {F = B} f)) (sym (isContrMor d .fst .snd c f))) ∙ sym (E .⋆Assoc _ _ _) - F-seq3 : (F : Functor D E) {x y z w : D .ob} + F-seq3 : (F : Functor D E) {x y z w : D .ob} → {f : D [ x , y ]}{g : D [ y , z ]}{h : D [ z , w ]} - → F .F-hom (f ⋆⟨ D ⟩ g ⋆⟨ D ⟩ h) ≡ F .F-hom f ⋆⟨ E ⟩ F .F-hom g ⋆⟨ E ⟩ F .F-hom h - F-seq3 F = F .F-seq _ _ ∙ cong (λ x → x ⋆⟨ E ⟩ _) (F .F-seq _ _) + → F .F-hom (f ⋆⟨ D ⟩ g ⋆⟨ D ⟩ h) ≡ F .F-hom f ⋆⟨ E ⟩ F .F-hom g ⋆⟨ E ⟩ F .F-hom h + F-seq3 F = F .F-seq _ _ ∙ cong (λ x → x ⋆⟨ E ⟩ _) (F .F-seq _ _) ext : NatTrans A B ext .N-ob d = isContrMor d .fst .fst ext .N-hom {x = d} {y = d'} f = Prop.rec2 (E .isSetHom _ _) (λ (c , h) (c' , h') → Prop.rec (E .isSetHom _ _) (λ (k , p) → - (λ i → A .F-hom f ⋆⟨ E ⟩ mor-eq d' c' h' i) - ∙ cong (λ x → A .F-hom f ⋆⟨ E ⟩ x) (E .⋆Assoc _ _ _) + (λ i → A .F-hom f ⋆⟨ E ⟩ mor-eq d' c' h' i) + ∙ cong (λ x → A .F-hom f ⋆⟨ E ⟩ x) (E .⋆Assoc _ _ _) ∙ sym (E .⋆Assoc _ _ _) ∙ sym (E .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (sym (E .⋆IdL _)) ∙ cong (λ x → x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (sym (F-Iso {F = A} h .snd .sec)) ∙ cong (λ x → x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (E .⋆Assoc _ _ _) ∙ cong (λ x → _ ⋆⟨ E ⟩ x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (sym (E .⋆Assoc _ _ _)) ∙ cong (λ x → _ ⋆⟨ E ⟩ x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (sym (F-seq3 A)) - ∙ cong (λ x → A .F-hom (invIso h .fst) ⋆⟨ E ⟩ x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (sym (cong (A .F-hom) p)) + ∙ cong (λ x → A .F-hom (invIso h .fst) ⋆⟨ E ⟩ x ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ _) (sym (cong (A .F-hom) p)) ∙ cong (λ x → x ⋆⟨ E ⟩ _) (E .⋆Assoc _ _ _) ∙ cong (λ x → _ ⋆⟨ E ⟩ x ⋆⟨ E ⟩ _) (α .N-hom k) ∙ cong (λ x → x ⋆⟨ E ⟩ _) (sym (E .⋆Assoc _ _ _)) ∙ E .⋆Assoc _ _ _ - ∙ cong (λ x → _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ (x ⋆⟨ E ⟩ B .F-hom (h' .fst))) (cong (B .F-hom) p) - ∙ cong (λ x → _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ (x ⋆⟨ E ⟩ B .F-hom (h' .fst))) (F-seq3 B) + ∙ cong (λ x → _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ (x ⋆⟨ E ⟩ B .F-hom (h' .fst))) (cong (B .F-hom) p) + ∙ cong (λ x → _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ (x ⋆⟨ E ⟩ B .F-hom (h' .fst))) (F-seq3 B) ∙ cong (λ x → _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ x) (E .⋆Assoc _ _ _) ∙ cong (λ x → _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ (_ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ x)) (F-Iso {F = B} h' .snd .sec) ∙ cong (λ x → _ ⋆⟨ E ⟩ _ ⋆⟨ E ⟩ x) (E .⋆IdR _) ∙ sym (E .⋆Assoc _ _ _) - ∙ (λ i → mor-eq d c h (~ i) ⋆⟨ E ⟩ B .F-hom f)) + ∙ (λ i → mor-eq d c h (~ i) ⋆⟨ E ⟩ B .F-hom f)) (full _ _ (h .fst ⋆⟨ D ⟩ f ⋆⟨ D ⟩ h' .snd .inv))) (surj d) (surj d') - ext≡ : precomposeF E F .F-hom ext ≡ α + ext≡ : precomposeF E F .F-hom ext ≡ α ext≡ = makeNatTransPath (λ i c → (mor-eq _ c idCatIso - ∙ (λ i → A .F-id i ⋆⟨ E ⟩ α .N-ob c ⋆⟨ E ⟩ B .F-id i) + ∙ (λ i → A .F-id i ⋆⟨ E ⟩ α .N-ob c ⋆⟨ E ⟩ B .F-id i) ∙ E .⋆IdR _ ∙ E .⋆IdL _) i) -- As a corollary, if F is essential surjective and full, (- ∘ F) is fully faithfull. - isEssSurj+Full→isFullyFaithfulPrecomp : isEssentiallySurj F → isFull F → isFullyFaithful (precomposeF E F) + isEssSurj+Full→isFullyFaithfulPrecomp : isEssentiallySurj F → isFull F → isFullyFaithful (precomposeF E F) isEssSurj+Full→isFullyFaithfulPrecomp surj full = isFull+Faithful→isFullyFaithful {F = precomposeF E F} (isEssSurj+Full→isFullPrecomp surj full) (isEssSurj→isFaithfulPrecomp surj) @@ -144,11 +144,11 @@ module _ {ℓC ℓC' ℓD ℓD' ℓE ℓE'} {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} {E : Category ℓE ℓE'} (isUnivE : isUnivalent E) - (F : Functor C D) + (F : Functor C D) where open Category - open Functor + open Functor open NatTrans open isIso open isWeakEquivalence @@ -157,42 +157,42 @@ -- If F is weak equivalence and the target category is univalent, (- ∘ F) is essentially surjective. - isWeakEquiv→isEssSurjPrecomp : isWeakEquivalence F → isEssentiallySurj (precomposeF E F) + isWeakEquiv→isEssSurjPrecomp : isWeakEquivalence F → isEssentiallySurj (precomposeF E F) isWeakEquiv→isEssSurjPrecomp w-equiv G = ∣ Ext , Ext≃ ∣₁ where Obj : (d : D .ob) → Type _ Obj d = Σ[ e ∈ E .ob ] - Σ[ k ∈ ((c : C .ob)(h : CatIso D (F .F-ob c) d) → CatIso E (G .F-ob c) e) ] - ((c c' : C .ob)(h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d) + Σ[ k ∈ ((c : C .ob)(h : CatIso D (F .F-ob c) d) → CatIso E (G .F-ob c) e) ] + ((c c' : C .ob)(h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d) → (f : C [ c , c' ]) - → F .F-hom f ⋆⟨ D ⟩ h' .fst ≡ h .fst - → G .F-hom f ⋆⟨ E ⟩ k c' h' .fst ≡ k c h .fst) + → F .F-hom f ⋆⟨ D ⟩ h' .fst ≡ h .fst + → G .F-hom f ⋆⟨ E ⟩ k c' h' .fst ≡ k c h .fst) - module _ {d} {c c'} (h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d) where + module _ {d} {c c'} (h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d) where liftH : CatIso C c c' liftH = liftIso {F = F} (w-equiv .fullfaith) (⋆Iso h (invIso h')) - liftH-eq' : F .F-hom (liftH .fst) ≡ h .fst ⋆⟨ D ⟩ h' .snd .inv + liftH-eq' : F .F-hom (liftH .fst) ≡ h .fst ⋆⟨ D ⟩ h' .snd .inv liftH-eq' = cong fst (liftIso≡ {F = F} (w-equiv .fullfaith) (⋆Iso h (invIso h'))) - liftH-eq : F .F-hom (liftH .fst) ⋆⟨ D ⟩ h' .fst ≡ h .fst + liftH-eq : F .F-hom (liftH .fst) ⋆⟨ D ⟩ h' .fst ≡ h .fst liftH-eq = cong (λ x → x ⋆⟨ D ⟩ _) liftH-eq' ∙ D .⋆Assoc _ _ _ ∙ cong (λ x → _ ⋆⟨ D ⟩ x) (h' .snd .sec) ∙ D .⋆IdR _ - module _ (d : D .ob)(c₀ : C .ob)(h₀ : CatIso D (F .F-ob c₀) d) where + module _ (d : D .ob)(c₀ : C .ob)(h₀ : CatIso D (F .F-ob c₀) d) where isContrObj' : isContr (Obj d) - isContrObj' .fst .fst = G .F-ob c₀ + isContrObj' .fst .fst = G .F-ob c₀ isContrObj' .fst .snd .fst c h = F-Iso {F = G} (liftH h h₀) - isContrObj' .fst .snd .snd c c' h h' f p = sym (G .F-seq _ _) ∙ cong (G .F-hom) g-path + isContrObj' .fst .snd .snd c c' h h' f p = sym (G .F-seq _ _) ∙ cong (G .F-hom) g-path where g-path : f ⋆⟨ C ⟩ liftH h' h₀ .fst ≡ liftH h h₀ .fst g-path = isFullyFaithful→Faithful {F = F} (w-equiv .fullfaith) _ _ _ _ - (F .F-seq _ _ + (F .F-seq _ _ ∙ cong (λ x → _ ⋆⟨ D ⟩ x) (liftH-eq' h' h₀) ∙ sym (D .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ D ⟩ invIso h₀ .fst) p @@ -207,7 +207,7 @@ isContrObj' .snd x@(e , k , coh) i .snd .snd = isProp→PathP (λ i → isPropΠ6 (λ c c' h h' f _ → E .isSetHom - (G .F-hom f ⋆⟨ E ⟩ isContrObj' .snd x i .snd .fst c' h' .fst) + (G .F-hom f ⋆⟨ E ⟩ isContrObj' .snd x i .snd .fst c' h' .fst) (isContrObj' .snd x i .snd .fst c h .fst))) (isContrObj' .fst .snd .snd) coh i @@ -217,31 +217,31 @@ Ext-ob : D .ob → E .ob Ext-ob d = isContrObj d .fst .fst - k : {d : D .ob}{c : C .ob}(h : CatIso D (F .F-ob c) d) → CatIso E (G .F-ob c) (Ext-ob d) + k : {d : D .ob}{c : C .ob}(h : CatIso D (F .F-ob c) d) → CatIso E (G .F-ob c) (Ext-ob d) k = isContrObj _ .fst .snd .fst _ k-eq : {d : D .ob}{c c' : C .ob} - (h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d) + (h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d) → (f : C [ c , c' ]) - → F .F-hom f ⋆⟨ D ⟩ h' .fst ≡ h .fst - → G .F-hom f ⋆⟨ E ⟩ k h' .fst ≡ k h .fst + → F .F-hom f ⋆⟨ D ⟩ h' .fst ≡ h .fst + → G .F-hom f ⋆⟨ E ⟩ k h' .fst ≡ k h .fst k-eq = isContrObj _ .fst .snd .snd _ _ Mor : (d d' : D .ob)(f : D [ d , d' ]) → Type _ Mor d d' f = Σ[ g ∈ E [ Ext-ob d , Ext-ob d' ] ] - ((c c' : C .ob)(h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d') + ((c c' : C .ob)(h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d') → (l : C [ c , c' ]) - → F .F-hom l ⋆⟨ D ⟩ h' .fst ≡ h .fst ⋆⟨ D ⟩ f - → G .F-hom l ⋆⟨ E ⟩ k h' .fst ≡ k h .fst ⋆⟨ E ⟩ g) + → F .F-hom l ⋆⟨ D ⟩ h' .fst ≡ h .fst ⋆⟨ D ⟩ f + → G .F-hom l ⋆⟨ E ⟩ k h' .fst ≡ k h .fst ⋆⟨ E ⟩ g) module _ (d d' : D .ob)(f : D [ d , d' ]) - (c₀ : C .ob)(h₀ : CatIso D (F .F-ob c₀) d ) - (c₁ : C .ob)(h₁ : CatIso D (F .F-ob c₁) d') - (l₀ : C [ c₀ , c₁ ])(p₀ : F .F-hom l₀ ⋆⟨ D ⟩ h₁ .fst ≡ h₀ .fst ⋆⟨ D ⟩ f) + (c₀ : C .ob)(h₀ : CatIso D (F .F-ob c₀) d ) + (c₁ : C .ob)(h₁ : CatIso D (F .F-ob c₁) d') + (l₀ : C [ c₀ , c₁ ])(p₀ : F .F-hom l₀ ⋆⟨ D ⟩ h₁ .fst ≡ h₀ .fst ⋆⟨ D ⟩ f) where - g₀ = k h₀ .snd .inv ⋆⟨ E ⟩ G .F-hom l₀ ⋆⟨ E ⟩ k h₁ .fst + g₀ = k h₀ .snd .inv ⋆⟨ E ⟩ G .F-hom l₀ ⋆⟨ E ⟩ k h₁ .fst isContrMor' : isContr (Mor d d' f) isContrMor' .fst .fst = g₀ @@ -249,7 +249,7 @@ cong (λ x → _ ⋆⟨ E ⟩ x) (⋆InvLMove (F-Iso {F = G} m') (k-eq h₁ h' _ m'-eq)) ∙ sym (E .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ E ⟩ k h₁ .fst) Gm-path' - ∙ cong (λ x → x ⋆⟨ E ⟩ G .F-hom l₀ ⋆⟨ E ⟩ k h₁ .fst) m-eq' + ∙ cong (λ x → x ⋆⟨ E ⟩ G .F-hom l₀ ⋆⟨ E ⟩ k h₁ .fst) m-eq' ∙ E .⋆Assoc _ _ _ ∙ E .⋆Assoc _ _ _ ∙ cong (λ x → k h .fst ⋆⟨ E ⟩ x) (sym (E .⋆Assoc _ _ _)) where @@ -258,13 +258,13 @@ m' = liftH h₁ h' m'-eq = liftH-eq h₁ h' - m-eq' : G .F-hom (m .snd .inv) ≡ k h .fst ⋆⟨ E ⟩ k h₀ .snd .inv + m-eq' : G .F-hom (m .snd .inv) ≡ k h .fst ⋆⟨ E ⟩ k h₀ .snd .inv m-eq' = ⋆InvRMove (k h₀) (sym (⋆InvLMove (F-Iso {F = G} m) (k-eq h₀ h _ m-eq))) - Fm-path : F .F-hom (l₀ ⋆⟨ C ⟩ m' .fst) ≡ F .F-hom (m .fst ⋆⟨ C ⟩ l) + Fm-path : F .F-hom (l₀ ⋆⟨ C ⟩ m' .fst) ≡ F .F-hom (m .fst ⋆⟨ C ⟩ l) Fm-path = - F .F-seq _ _ - ∙ cong (λ x → F .F-hom l₀ ⋆⟨ D ⟩ x) (liftH-eq' h₁ h') + F .F-seq _ _ + ∙ cong (λ x → F .F-hom l₀ ⋆⟨ D ⟩ x) (liftH-eq' h₁ h') ∙ sym (D .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ D ⟩ _) p₀ ∙ cong (λ x → x ⋆⟨ D ⟩ _ ⋆⟨ D ⟩ _) (sym (D .⋆IdR _)) @@ -273,15 +273,15 @@ ∙ cong (λ x → x ⋆⟨ D ⟩ _) (D .⋆Assoc _ _ _) ∙ D .⋆Assoc _ _ _ ∙ (λ i → ⋆InvRMove h m-eq (~ i) ⋆⟨ D ⟩ ⋆InvRMove h' p (~ i)) - ∙ sym (F .F-seq _ _) + ∙ sym (F .F-seq _ _) m-path : l₀ ⋆⟨ C ⟩ m' .fst ≡ m .fst ⋆⟨ C ⟩ l m-path = isFullyFaithful→Faithful {F = F} (w-equiv .fullfaith) _ _ _ _ Fm-path - Gm-path : G .F-hom l₀ ⋆⟨ E ⟩ G .F-hom (m' .fst) ≡ G .F-hom (m .fst) ⋆⟨ E ⟩ G .F-hom l - Gm-path = sym (G .F-seq _ _) ∙ cong (G .F-hom) m-path ∙ G .F-seq _ _ + Gm-path : G .F-hom l₀ ⋆⟨ E ⟩ G .F-hom (m' .fst) ≡ G .F-hom (m .fst) ⋆⟨ E ⟩ G .F-hom l + Gm-path = sym (G .F-seq _ _) ∙ cong (G .F-hom) m-path ∙ G .F-seq _ _ - Gm-path' : G .F-hom l ⋆⟨ E ⟩ G .F-hom (m' .snd .inv) ≡ G .F-hom (m .snd .inv) ⋆⟨ E ⟩ G .F-hom l₀ + Gm-path' : G .F-hom l ⋆⟨ E ⟩ G .F-hom (m' .snd .inv) ≡ G .F-hom (m .snd .inv) ⋆⟨ E ⟩ G .F-hom l₀ Gm-path' = ⋆InvLMove (F-Iso {F = G} m) (sym (⋆InvRMove (F-Iso {F = G} m') Gm-path ∙ E .⋆Assoc _ _ _)) isContrMor' .snd (g₁ , coh₁) i .fst = @@ -290,18 +290,18 @@ isContrMor' .snd x@(g₁ , coh₁) i .snd = isProp→PathP (λ i → isPropΠ6 (λ c c' h h' l _ → E .isSetHom - (G .F-hom l ⋆⟨ E ⟩ k h' .fst) + (G .F-hom l ⋆⟨ E ⟩ k h' .fst) (k h .fst ⋆⟨ E ⟩ isContrMor' .snd x i .fst))) (isContrMor' .fst .snd) coh₁ i module _ {c c'} {d d'} - (f : D [ d , d' ])(h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d') + (f : D [ d , d' ])(h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d') where liftL : C [ c , c' ] liftL = invEq (_ , w-equiv .fullfaith _ _) (h .fst ⋆⟨ D ⟩ f ⋆⟨ D ⟩ h' .snd .inv) - liftL-eq : F .F-hom liftL ⋆⟨ D ⟩ h' .fst ≡ h .fst ⋆⟨ D ⟩ f + liftL-eq : F .F-hom liftL ⋆⟨ D ⟩ h' .fst ≡ h .fst ⋆⟨ D ⟩ f liftL-eq = sym (⋆InvRMove (invIso h') (sym (secEq (_ , w-equiv .fullfaith _ _) (h .fst ⋆⟨ D ⟩ f ⋆⟨ D ⟩ h' .snd .inv)))) @@ -317,29 +317,29 @@ liftL⋆ : ∀ {c c' c''} {d d' d''} (f : D [ d , d' ])(g : D [ d' , d'' ]) - (h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d')(h'' : CatIso D (F .F-ob c'') d'') + (h : CatIso D (F .F-ob c) d)(h' : CatIso D (F .F-ob c') d')(h'' : CatIso D (F .F-ob c'') d'') → liftL (f ⋆⟨ D ⟩ g) h h'' ≡ liftL f h h' ⋆⟨ C ⟩ liftL g h' h'' liftL⋆ f g h h' h'' = isFullyFaithful→Faithful {F = F} (w-equiv .fullfaith) _ _ _ _ - (⋆CancelR h'' path ∙ sym (F .F-seq _ _)) + (⋆CancelR h'' path ∙ sym (F .F-seq _ _)) where path : _ path = liftL-eq (f ⋆⟨ D ⟩ g) h h'' ∙ sym (D .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ D ⟩ _) (sym (liftL-eq f h h')) ∙ D .⋆Assoc _ _ _ - ∙ cong (λ x → F .F-hom (liftL f h h') ⋆⟨ D ⟩ x) (sym (liftL-eq g h' h'')) + ∙ cong (λ x → F .F-hom (liftL f h h') ⋆⟨ D ⟩ x) (sym (liftL-eq g h' h'')) ∙ sym (D .⋆Assoc _ _ _) - Ext : Functor D E - Ext .F-ob = Ext-ob - Ext .F-hom = Ext-hom - Ext .F-id {x = d} = Prop.rec (E .isSetHom _ _) + Ext : Functor D E + Ext .F-ob = Ext-ob + Ext .F-hom = Ext-hom + Ext .F-id {x = d} = Prop.rec (E .isSetHom _ _) (λ (c , h) → let r = isContrMor _ _ (D .id {x = d}) .fst .snd _ _ h h (C .id) - (cong (λ x → x ⋆⟨ D ⟩ _) (F .F-id) ∙ D .⋆IdL _ ∙ sym (D .⋆IdR _)) - in ⋆CancelL (k h) (sym r ∙ cong (λ x → x ⋆⟨ E ⟩ (k h .fst)) (G .F-id) ∙ E .⋆IdL _ ∙ sym (E .⋆IdR _))) + (cong (λ x → x ⋆⟨ D ⟩ _) (F .F-id) ∙ D .⋆IdL _ ∙ sym (D .⋆IdR _)) + in ⋆CancelL (k h) (sym r ∙ cong (λ x → x ⋆⟨ E ⟩ (k h .fst)) (G .F-id) ∙ E .⋆IdL _ ∙ sym (E .⋆IdR _))) (w-equiv .esssurj d) - Ext .F-seq {x = a} {y = b} {z = c} f g = + Ext .F-seq {x = a} {y = b} {z = c} f g = Prop.rec3 (E .isSetHom _ _) (λ (_ , ha) (_ , hb) (_ , hc) → let rf = isContrMor _ _ _ .fst .snd _ _ ha hb (liftL f ha hb) (liftL-eq f ha hb) @@ -347,62 +347,62 @@ rfg = isContrMor _ _ _ .fst .snd _ _ ha hc (liftL (f ⋆⟨ D ⟩ g) ha hc) (liftL-eq (f ⋆⟨ D ⟩ g) ha hc) in ⋆CancelL (k ha) (sym rfg - ∙ cong (λ x → x ⋆⟨ E ⟩ k hc .fst) (cong (G .F-hom) (liftL⋆ f g ha hb hc) ∙ G .F-seq _ _) + ∙ cong (λ x → x ⋆⟨ E ⟩ k hc .fst) (cong (G .F-hom) (liftL⋆ f g ha hb hc) ∙ G .F-seq _ _) ∙ E .⋆Assoc _ _ _ - ∙ cong (λ x → G .F-hom _ ⋆⟨ E ⟩ x) rg + ∙ cong (λ x → G .F-hom _ ⋆⟨ E ⟩ x) rg ∙ sym (E .⋆Assoc _ _ _) ∙ cong (λ x → x ⋆⟨ E ⟩ Ext-hom g) rf ∙ E .⋆Assoc _ _ _)) (w-equiv .esssurj a) (w-equiv .esssurj b) (w-equiv .esssurj c) - objFc : (c : C .ob) → Obj (F .F-ob c) - objFc c₀ .fst = G .F-ob c₀ + objFc : (c : C .ob) → Obj (F .F-ob c) + objFc c₀ .fst = G .F-ob c₀ objFc c₀ .snd .fst c h = F-Iso {F = G} (liftIso {F = F} (w-equiv .fullfaith) h) - objFc c₀ .snd .snd c c' h h' f p = sym (G .F-seq _ _) - ∙ cong (G .F-hom) (isFullyFaithful→Faithful {F = F} (w-equiv .fullfaith) _ _ _ _ path) + objFc c₀ .snd .snd c c' h h' f p = sym (G .F-seq _ _) + ∙ cong (G .F-hom) (isFullyFaithful→Faithful {F = F} (w-equiv .fullfaith) _ _ _ _ path) where path : _ path = - F .F-seq _ _ + F .F-seq _ _ ∙ cong (λ x → _ ⋆⟨ D ⟩ x) (cong fst (liftIso≡ {F = F} (w-equiv .fullfaith) h')) ∙ p ∙ sym (cong fst (liftIso≡ {F = F} (w-equiv .fullfaith) h)) - Ext-ob≡ : (c : C .ob) → Ext-ob (F .F-ob c) ≡ G .F-ob c + Ext-ob≡ : (c : C .ob) → Ext-ob (F .F-ob c) ≡ G .F-ob c Ext-ob≡ c₀ = cong fst (isContrObj _ .snd (objFc c₀)) Ext-hom≡ : {c c' : C .ob}(f : C [ c , c' ]) - → PathP (λ i → E [ Ext-ob≡ c i , Ext-ob≡ c' i ]) (Ext-hom (F .F-hom f)) (G .F-hom f) + → PathP (λ i → E [ Ext-ob≡ c i , Ext-ob≡ c' i ]) (Ext-hom (F .F-hom f)) (G .F-hom f) Ext-hom≡ {c = c} {c' = c'} f i = hcomp (λ j → λ { (i = i0) → isContrMor _ _ _ .snd morFf (~ j) .fst - ; (i = i1) → G .F-hom f }) + ; (i = i1) → G .F-hom f }) (transpGf-filler (~ i)) where - transpGf-filler = transport-filler (λ i → E [ Ext-ob≡ c (~ i) , Ext-ob≡ c' (~ i) ]) (G .F-hom f) + transpGf-filler = transport-filler (λ i → E [ Ext-ob≡ c (~ i) , Ext-ob≡ c' (~ i) ]) (G .F-hom f) - morFf : Mor _ _ (F .F-hom f) + morFf : Mor _ _ (F .F-hom f) morFf .fst = transpGf-filler i1 morFf .snd c c' h h' l p = - transport (λ i → G .F-hom l ⋆⟨ E ⟩ isContrObj _ .snd (objFc _) (~ i) .snd .fst c' h' .fst + transport (λ i → G .F-hom l ⋆⟨ E ⟩ isContrObj _ .snd (objFc _) (~ i) .snd .fst c' h' .fst ≡ isContrObj _ .snd (objFc _) (~ i) .snd .fst c h .fst ⋆⟨ E ⟩ transpGf-filler i) G-path where F-path : _ F-path = - F .F-seq _ _ + F .F-seq _ _ ∙ cong (λ x → _ ⋆⟨ D ⟩ x) (cong fst (liftIso≡ {F = F} (w-equiv .fullfaith) h')) ∙ p ∙ cong (λ x → x ⋆⟨ D ⟩ _) (sym (cong fst (liftIso≡ {F = F} (w-equiv .fullfaith) h))) - ∙ sym (F .F-seq _ _) + ∙ sym (F .F-seq _ _) G-path : _ G-path = - sym (G .F-seq _ _) - ∙ cong (G .F-hom) (isFullyFaithful→Faithful {F = F} (w-equiv .fullfaith) _ _ _ _ F-path) - ∙ G .F-seq _ _ + sym (G .F-seq _ _) + ∙ cong (G .F-hom) (isFullyFaithful→Faithful {F = F} (w-equiv .fullfaith) _ _ _ _ F-path) + ∙ G .F-seq _ _ - Ext≡ : precomposeF E F .F-ob Ext ≡ G - Ext≡ = Functor≡ Ext-ob≡ Ext-hom≡ + Ext≡ : precomposeF E F .F-ob Ext ≡ G + Ext≡ = Functor≡ Ext-ob≡ Ext-hom≡ - Ext≃ : CatIso _ (precomposeF E F .F-ob Ext) G + Ext≃ : CatIso _ (precomposeF E F .F-ob Ext) G Ext≃ = NatIso→FUNCTORIso _ _ (pathToNatIso Ext≡) diff --git a/Cubical.Categories.Functor.Properties.html b/Cubical.Categories.Functor.Properties.html index c75113e329..5cd7e6e203 100644 --- a/Cubical.Categories.Functor.Properties.html +++ b/Cubical.Categories.Functor.Properties.html @@ -25,134 +25,134 @@ B C D E : Category ℓ ℓ' open Category -open Functor +open Functor -F-assoc : {F : Functor B C} {G : Functor C D} {H : Functor D E} - → H ∘F (G ∘F F) ≡ (H ∘F G) ∘F F -F-assoc = Functor≡ (λ _ → refl) (λ _ → refl) +F-assoc : {F : Functor B C} {G : Functor C D} {H : Functor D E} + → H ∘F (G ∘F F) ≡ (H ∘F G) ∘F F +F-assoc = Functor≡ (λ _ → refl) (λ _ → refl) -- Results about functors -module _ {F : Functor C D} where +module _ {F : Functor C D} where -- the identity is the identity - F-lUnit : F ∘F 𝟙⟨ C ⟩ ≡ F - F-lUnit i .F-ob x = F ⟅ x ⟆ - F-lUnit i .F-hom f = F ⟪ f ⟫ - F-lUnit i .F-id {x} = lUnit (F .F-id) (~ i) - F-lUnit i .F-seq f g = lUnit (F .F-seq f g) (~ i) - - F-rUnit : 𝟙⟨ D ⟩ ∘F F ≡ F - F-rUnit i .F-ob x = F ⟅ x ⟆ - F-rUnit i .F-hom f = F ⟪ f ⟫ - F-rUnit i .F-id {x} = rUnit (F .F-id) (~ i) - F-rUnit i .F-seq f g = rUnit (F .F-seq f g) (~ i) + F-lUnit : F ∘F 𝟙⟨ C ⟩ ≡ F + F-lUnit i .F-ob x = F ⟅ x ⟆ + F-lUnit i .F-hom f = F ⟪ f ⟫ + F-lUnit i .F-id {x} = lUnit (F .F-id) (~ i) + F-lUnit i .F-seq f g = lUnit (F .F-seq f g) (~ i) + + F-rUnit : 𝟙⟨ D ⟩ ∘F F ≡ F + F-rUnit i .F-ob x = F ⟅ x ⟆ + F-rUnit i .F-hom f = F ⟪ f ⟫ + F-rUnit i .F-id {x} = rUnit (F .F-id) (~ i) + F-rUnit i .F-seq f g = rUnit (F .F-seq f g) (~ i) -- functors preserve commutative diagrams (specificallysqures here) preserveCommF : ∀ {x y z w} {f : C [ x , y ]} {g : C [ y , w ]} {h : C [ x , z ]} {k : C [ z , w ]} → f ⋆⟨ C ⟩ g ≡ h ⋆⟨ C ⟩ k - → (F ⟪ f ⟫) ⋆⟨ D ⟩ (F ⟪ g ⟫) ≡ (F ⟪ h ⟫) ⋆⟨ D ⟩ (F ⟪ k ⟫) + → (F ⟪ f ⟫) ⋆⟨ D ⟩ (F ⟪ g ⟫) ≡ (F ⟪ h ⟫) ⋆⟨ D ⟩ (F ⟪ k ⟫) preserveCommF {f = f} {g = g} {h = h} {k = k} eq - = (F ⟪ f ⟫) ⋆⟨ D ⟩ (F ⟪ g ⟫) - ≡⟨ sym (F .F-seq _ _) ⟩ - F ⟪ f ⋆⟨ C ⟩ g ⟫ - ≡⟨ cong (F ⟪_⟫) eq ⟩ - F ⟪ h ⋆⟨ C ⟩ k ⟫ - ≡⟨ F .F-seq _ _ ⟩ - (F ⟪ h ⟫) ⋆⟨ D ⟩ (F ⟪ k ⟫) + = (F ⟪ f ⟫) ⋆⟨ D ⟩ (F ⟪ g ⟫) + ≡⟨ sym (F .F-seq _ _) ⟩ + F ⟪ f ⋆⟨ C ⟩ g ⟫ + ≡⟨ cong (F ⟪_⟫) eq ⟩ + F ⟪ h ⋆⟨ C ⟩ k ⟫ + ≡⟨ F .F-seq _ _ ⟩ + (F ⟪ h ⟫) ⋆⟨ D ⟩ (F ⟪ k ⟫) ∎ -- functors preserve isomorphisms - preserveIsosF : ∀ {x y} → CatIso C x y → CatIso D (F ⟅ x ⟆) (F ⟅ y ⟆) + preserveIsosF : ∀ {x y} → CatIso C x y → CatIso D (F ⟅ x ⟆) (F ⟅ y ⟆) preserveIsosF {x} {y} (f , isiso f⁻¹ sec' ret') = catiso g g⁻¹ -- sec ( (g⁻¹ ⋆⟨ D ⟩ g) - ≡⟨ sym (F .F-seq f⁻¹ f) ⟩ - F ⟪ f⁻¹ ⋆⟨ C ⟩ f ⟫ - ≡⟨ cong (F .F-hom) sec' ⟩ - F ⟪ C .id ⟫ - ≡⟨ F .F-id ⟩ + ≡⟨ sym (F .F-seq f⁻¹ f) ⟩ + F ⟪ f⁻¹ ⋆⟨ C ⟩ f ⟫ + ≡⟨ cong (F .F-hom) sec' ⟩ + F ⟪ C .id ⟫ + ≡⟨ F .F-id ⟩ D .id ∎ ) -- ret ( (g ⋆⟨ D ⟩ g⁻¹) - ≡⟨ sym (F .F-seq f f⁻¹) ⟩ - F ⟪ f ⋆⟨ C ⟩ f⁻¹ ⟫ - ≡⟨ cong (F .F-hom) ret' ⟩ - F ⟪ C .id ⟫ - ≡⟨ F .F-id ⟩ + ≡⟨ sym (F .F-seq f f⁻¹) ⟩ + F ⟪ f ⋆⟨ C ⟩ f⁻¹ ⟫ + ≡⟨ cong (F .F-hom) ret' ⟩ + F ⟪ C .id ⟫ + ≡⟨ F .F-id ⟩ D .id ∎ ) where x' : D .ob - x' = F ⟅ x ⟆ + x' = F ⟅ x ⟆ y' : D .ob - y' = F ⟅ y ⟆ + y' = F ⟅ y ⟆ g : D [ x' , y' ] - g = F ⟪ f ⟫ + g = F ⟪ f ⟫ g⁻¹ : D [ y' , x' ] - g⁻¹ = F ⟪ f⁻¹ ⟫ + g⁻¹ = F ⟪ f⁻¹ ⟫ -- hacky lemma helping with type inferences functorCongP : {x y v w : ob C} {p : x ≡ y} {q : v ≡ w} {f : C [ x , v ]} {g : C [ y , w ]} → PathP (λ i → C [ p i , q i ]) f g - → PathP (λ i → D [ F .F-ob (p i) , F. F-ob (q i) ]) (F .F-hom f) (F .F-hom g) - functorCongP r i = F .F-hom (r i) + → PathP (λ i → D [ F .F-ob (p i) , F. F-ob (q i) ]) (F .F-hom f) (F .F-hom g) + functorCongP r i = F .F-hom (r i) -isSetFunctor : isSet (D .ob) → isSet (Functor C D) +isSetFunctor : isSet (D .ob) → isSet (Functor C D) isSetFunctor {D = D} {C = C} isSet-D-ob F G p q = w where w : _ - F-ob (w i i₁) = isSetΠ (λ _ → isSet-D-ob) _ _ (cong F-ob p) (cong F-ob q) i i₁ - F-hom (w i i₁) z = + F-ob (w i i₁) = isSetΠ (λ _ → isSet-D-ob) _ _ (cong F-ob p) (cong F-ob q) i i₁ + F-hom (w i i₁) z = isSet→SquareP - (λ i i₁ → D .isSetHom {(F-ob (w i i₁) _)} {(F-ob (w i i₁) _)}) - (λ i₁ → F-hom (p i₁) z) (λ i₁ → F-hom (q i₁) z) refl refl i i₁ + (λ i i₁ → D .isSetHom {(F-ob (w i i₁) _)} {(F-ob (w i i₁) _)}) + (λ i₁ → F-hom (p i₁) z) (λ i₁ → F-hom (q i₁) z) refl refl i i₁ - F-id (w i i₁) = + F-id (w i i₁) = isSet→SquareP - (λ i i₁ → isProp→isSet (D .isSetHom (F-hom (w i i₁) _) (D .id))) - (λ i₁ → F-id (p i₁)) (λ i₁ → F-id (q i₁)) refl refl i i₁ + (λ i i₁ → isProp→isSet (D .isSetHom (F-hom (w i i₁) _) (D .id))) + (λ i₁ → F-id (p i₁)) (λ i₁ → F-id (q i₁)) refl refl i i₁ - F-seq (w i i₁) _ _ = + F-seq (w i i₁) _ _ = isSet→SquareP - (λ i i₁ → isProp→isSet (D .isSetHom (F-hom (w i i₁) _) ((F-hom (w i i₁) _) ⋆⟨ D ⟩ (F-hom (w i i₁) _)))) - (λ i₁ → F-seq (p i₁) _ _) (λ i₁ → F-seq (q i₁) _ _) refl refl i i₁ + (λ i i₁ → isProp→isSet (D .isSetHom (F-hom (w i i₁) _) ((F-hom (w i i₁) _) ⋆⟨ D ⟩ (F-hom (w i i₁) _)))) + (λ i₁ → F-seq (p i₁) _ _) (λ i₁ → F-seq (q i₁) _ _) refl refl i i₁ -- Conservative Functor, -- namely if a morphism f is mapped to an isomorphism, -- the morphism f is itself isomorphism. -isConservative : (F : Functor C D) → Type _ -isConservative {C = C} {D = D} F = {x y : C .ob}{f : C [ x , y ]} → isIso D (F .F-hom f) → isIso C f +isConservative : (F : Functor C D) → Type _ +isConservative {C = C} {D = D} F = {x y : C .ob}{f : C [ x , y ]} → isIso D (F .F-hom f) → isIso C f -- Fully-faithfulness of functors -module _ {F : Functor C D} where +module _ {F : Functor C D} where - isFullyFaithful→Full : isFullyFaithful F → isFull F + isFullyFaithful→Full : isFullyFaithful F → isFull F isFullyFaithful→Full fullfaith x y = isEquiv→isSurjection (fullfaith x y) - isFullyFaithful→Faithful : isFullyFaithful F → isFaithful F + isFullyFaithful→Faithful : isFullyFaithful F → isFaithful F isFullyFaithful→Faithful fullfaith x y = isEmbedding→Inj (isEquiv→isEmbedding (fullfaith x y)) - isFull+Faithful→isFullyFaithful : isFull F → isFaithful F → isFullyFaithful F + isFull+Faithful→isFullyFaithful : isFull F → isFaithful F → isFullyFaithful F isFull+Faithful→isFullyFaithful full faith x y = isEmbedding×isSurjection→isEquiv (injEmbedding (D .isSetHom) (faith x y _ _) , full x y) - isFaithful→reflectsMono : isFaithful F → {x y : C .ob} (f : C [ x , y ]) - → isMonic D (F ⟪ f ⟫) → isMonic C f + isFaithful→reflectsMono : isFaithful F → {x y : C .ob} (f : C [ x , y ]) + → isMonic D (F ⟪ f ⟫) → isMonic C f isFaithful→reflectsMono F-faithful f Ff-mon {a = a} {a' = a'} a⋆f≡a'⋆f = - let Fa⋆Ff≡Fa'⋆Ff = sym (F .F-seq a f) - ∙ cong (F ⟪_⟫) a⋆f≡a'⋆f - ∙ F .F-seq a' f + let Fa⋆Ff≡Fa'⋆Ff = sym (F .F-seq a f) + ∙ cong (F ⟪_⟫) a⋆f≡a'⋆f + ∙ F .F-seq a' f in F-faithful _ _ _ _ (Ff-mon Fa⋆Ff≡Fa'⋆Ff) @@ -160,46 +160,46 @@ open isIso - isFullyFaithful→Conservative : isFullyFaithful F → isConservative F + isFullyFaithful→Conservative : isFullyFaithful F → isConservative F isFullyFaithful→Conservative fullfaith {x = x} {y = y} {f = f} isoFf = w where w : isIso C f w .inv = invIsEq (fullfaith _ _) (isoFf .inv) w .sec = isFullyFaithful→Faithful fullfaith _ _ _ _ - (F .F-seq _ _ - ∙ (λ i → secIsEq (fullfaith _ _) (isoFf .inv) i ⋆⟨ D ⟩ F .F-hom f) + (F .F-seq _ _ + ∙ (λ i → secIsEq (fullfaith _ _) (isoFf .inv) i ⋆⟨ D ⟩ F .F-hom f) ∙ isoFf .sec - ∙ sym (F .F-id)) + ∙ sym (F .F-id)) w .ret = isFullyFaithful→Faithful fullfaith _ _ _ _ - (F .F-seq _ _ - ∙ (λ i → F .F-hom f ⋆⟨ D ⟩ secIsEq (fullfaith _ _) (isoFf .inv) i) + (F .F-seq _ _ + ∙ (λ i → F .F-hom f ⋆⟨ D ⟩ secIsEq (fullfaith _ _) (isoFf .inv) i) ∙ isoFf .ret - ∙ sym (F .F-id)) + ∙ sym (F .F-id)) -- Lifting isomorphism upwards a fully faithful functor - module _ (fullfaith : isFullyFaithful F) where + module _ (fullfaith : isFullyFaithful F) where - liftIso : {x y : C .ob} → CatIso D (F .F-ob x) (F .F-ob y) → CatIso C x y + liftIso : {x y : C .ob} → CatIso D (F .F-ob x) (F .F-ob y) → CatIso C x y liftIso f .fst = invEq (_ , fullfaith _ _) (f .fst) liftIso f .snd = isFullyFaithful→Conservative fullfaith (subst (isIso D) (sym (secEq (_ , fullfaith _ _) (f .fst))) (f .snd)) - liftIso≡ : {x y : C .ob} → (f : CatIso D (F .F-ob x) (F .F-ob y)) → F-Iso {F = F} (liftIso f) ≡ f + liftIso≡ : {x y : C .ob} → (f : CatIso D (F .F-ob x) (F .F-ob y)) → F-Iso {F = F} (liftIso f) ≡ f liftIso≡ f = CatIso≡ _ _ (secEq (_ , fullfaith _ _) (f .fst)) -- Functors inducing surjection on objects is essentially surjective -isSurj-ob→isSurj : {F : Functor C D} → isSurjection (F .F-ob) → isEssentiallySurj F +isSurj-ob→isSurj : {F : Functor C D} → isSurjection (F .F-ob) → isEssentiallySurj F isSurj-ob→isSurj surj y = Prop.map (λ (x , p) → x , pathToIso p) (surj y) -- Fully-faithful functors induce equivalence on isomorphisms -isFullyFaithful→isEquivF-Iso : {F : Functor C D} - → isFullyFaithful F → ∀ x y → isEquiv (F-Iso {F = F} {x = x} {y = y}) +isFullyFaithful→isEquivF-Iso : {F : Functor C D} + → isFullyFaithful F → ∀ x y → isEquiv (F-Iso {F = F} {x = x} {y = y}) isFullyFaithful→isEquivF-Iso {F = F} fullfaith x y = - Σ-cong-equiv-prop (_ , fullfaith x y) isPropIsIso isPropIsIso _ + Σ-cong-equiv-prop (_ , fullfaith x y) isPropIsIso isPropIsIso _ (λ f → isFullyFaithful→Conservative {F = F} fullfaith {f = f}) .snd @@ -213,21 +213,21 @@ -- Essentially surjective functor with univalent target induces surjection on objects - isSurj→isSurj-ob : {F : Functor C D} → isEssentiallySurj F → isSurjection (F .F-ob) + isSurj→isSurj-ob : {F : Functor C D} → isEssentiallySurj F → isSurjection (F .F-ob) isSurj→isSurj-ob surj y = Prop.map (λ (x , f) → x , CatIsoToPath f) (surj y) module _ (isUnivC : isUnivalent C) (isUnivD : isUnivalent D) - {F : Functor C D} + {F : Functor C D} where open isUnivalent -- Fully-faithful functor between univalent target induces embedding on objects - isFullyFaithful→isEmbd-ob : isFullyFaithful F → isEmbedding (F .F-ob) + isFullyFaithful→isEmbd-ob : isFullyFaithful F → isEmbedding (F .F-ob) isFullyFaithful→isEmbd-ob fullfaith x y = isEquiv[equivFunA≃B∘f]→isEquiv[f] _ (_ , isUnivD .univ _ _) (subst isEquiv (F-pathToIso-∘ {F = F}) diff --git a/Cubical.Categories.Functors.Constant.html b/Cubical.Categories.Functors.Constant.html index 87e5f8a0f9..6f9c39b198 100644 --- a/Cubical.Categories.Functors.Constant.html +++ b/Cubical.Categories.Functors.Constant.html @@ -12,11 +12,11 @@ ℓC ℓC' ℓD ℓD' : Level open Category -open Functor +open Functor -Constant : (C : Category ℓC ℓC') (D : Category ℓD ℓD') (d : ob D) → Functor C D -F-ob (Constant C D d) c = d -F-hom (Constant C D d) φ = id D -F-id (Constant C D d) = refl -F-seq (Constant C D d) φ χ = sym (⋆IdR D _) +Constant : (C : Category ℓC ℓC') (D : Category ℓD ℓD') (d : ob D) → Functor C D +F-ob (Constant C D d) c = d +F-hom (Constant C D d) φ = id D +F-id (Constant C D d) = refl +F-seq (Constant C D d) φ χ = sym (⋆IdR D _) \ No newline at end of file diff --git a/Cubical.Categories.Functors.HomFunctor.html b/Cubical.Categories.Functors.HomFunctor.html index 51640bd11b..7b6d64432c 100644 --- a/Cubical.Categories.Functors.HomFunctor.html +++ b/Cubical.Categories.Functors.HomFunctor.html @@ -15,14 +15,14 @@ module _ (C : Category ℓC ℓC') where - open Functor + open Functor open Category C - HomFunctor : Functor (C ^op ×C C) (SET ℓC') - F-ob HomFunctor (x , y) = Hom[ x , y ] , isSetHom - F-hom HomFunctor {x , y} {x' , y'} (φ , ψ) θ = ψ ∘ (θ ∘ φ) - F-id HomFunctor = funExt λ θ → ⋆IdR (id ⋆ θ) ∙ ⋆IdL θ - F-seq HomFunctor {x , y} {x' , y'} {x'' , y''} (φ , ψ) (φ' , ψ') = funExt λ θ → + HomFunctor : Functor (C ^op ×C C) (SET ℓC') + F-ob HomFunctor (x , y) = Hom[ x , y ] , isSetHom + F-hom HomFunctor {x , y} {x' , y'} (φ , ψ) θ = ψ ∘ (θ ∘ φ) + F-id HomFunctor = funExt λ θ → ⋆IdR (id ⋆ θ) ∙ ⋆IdL θ + F-seq HomFunctor {x , y} {x' , y'} {x'' , y''} (φ , ψ) (φ' , ψ') = funExt λ θ → ((φ' ⋆ φ) ⋆ θ) ⋆ (ψ ⋆ ψ') ≡⟨ sym (⋆Assoc ((φ' ⋆ φ) ⋆ θ) ψ ψ') ⟩ (((φ' ⋆ φ) ⋆ θ) ⋆ ψ) ⋆ ψ' diff --git a/Cubical.Categories.Instances.Categories.html b/Cubical.Categories.Instances.Categories.html index 632a9e140a..d5a9054829 100644 --- a/Cubical.Categories.Instances.Categories.html +++ b/Cubical.Categories.Instances.Categories.html @@ -16,9 +16,9 @@ CatPrecategory : Precategory (ℓ-suc (ℓ-max ℓ ℓ')) (ℓ-max ℓ ℓ') CatPrecategory .ob = Category ℓ ℓ' - CatPrecategory .Hom[_,_] = Functor - CatPrecategory .id = 𝟙⟨ _ ⟩ - CatPrecategory ._⋆_ G H = H ∘F G + CatPrecategory .Hom[_,_] = Functor + CatPrecategory .id = 𝟙⟨ _ ⟩ + CatPrecategory ._⋆_ G H = H ∘F G CatPrecategory .⋆IdL _ = F-lUnit CatPrecategory .⋆IdR _ = F-rUnit CatPrecategory .⋆Assoc _ _ _ = F-assoc diff --git a/Cubical.Categories.Instances.CommAlgebras.html b/Cubical.Categories.Instances.CommAlgebras.html index 0822f7b468..5656bc818e 100644 --- a/Cubical.Categories.Instances.CommAlgebras.html +++ b/Cubical.Categories.Instances.CommAlgebras.html @@ -56,18 +56,18 @@ snd (snd TerminalCommAlgebra A) f = AlgebraHom≡ (funExt (λ _ → refl)) -- the forgetful functor into CommRings and SET - open Functor - ForgetfulCommAlgebra→CommRing : Functor (CommAlgebrasCategory {ℓ' = ℓ'}) CommRingsCategory - F-ob ForgetfulCommAlgebra→CommRing = CommAlgebra→CommRing {R = R} - F-hom ForgetfulCommAlgebra→CommRing = CommAlgebraHom→CommRingHom _ _ - F-id ForgetfulCommAlgebra→CommRing = RingHom≡ refl - F-seq ForgetfulCommAlgebra→CommRing _ _ = RingHom≡ refl - - ForgetfulCommAlgebra→Set : Functor (CommAlgebrasCategory {ℓ' = ℓ'}) (SET ℓ') - F-ob ForgetfulCommAlgebra→Set A = A .fst , A .snd .CommAlgebraStr.is-set - F-hom ForgetfulCommAlgebra→Set = fst - F-id ForgetfulCommAlgebra→Set = refl - F-seq ForgetfulCommAlgebra→Set _ _ = refl + open Functor + ForgetfulCommAlgebra→CommRing : Functor (CommAlgebrasCategory {ℓ' = ℓ'}) CommRingsCategory + F-ob ForgetfulCommAlgebra→CommRing = CommAlgebra→CommRing {R = R} + F-hom ForgetfulCommAlgebra→CommRing = CommAlgebraHom→CommRingHom _ _ + F-id ForgetfulCommAlgebra→CommRing = RingHom≡ refl + F-seq ForgetfulCommAlgebra→CommRing _ _ = RingHom≡ refl + + ForgetfulCommAlgebra→Set : Functor (CommAlgebrasCategory {ℓ' = ℓ'}) (SET ℓ') + F-ob ForgetfulCommAlgebra→Set A = A .fst , A .snd .CommAlgebraStr.is-set + F-hom ForgetfulCommAlgebra→Set = fst + F-id ForgetfulCommAlgebra→Set = refl + F-seq ForgetfulCommAlgebra→Set _ _ = refl -- general limits module _ {ℓJ ℓJ' : Level} where @@ -87,47 +87,47 @@ open IsAlgebraHom LimitsCommAlgebrasCategory : Limits {ℓ-max ℓ ℓJ} {ℓJ'} (CommAlgebrasCategory {ℓ' = theℓ}) - fst (lim (LimitsCommAlgebrasCategory J D)) = lim (completeSET J (ffA→SET ∘F D)) .fst + fst (lim (LimitsCommAlgebrasCategory J D)) = lim (completeSET J (ffA→SET ∘F D)) .fst 0a (snd (lim (LimitsCommAlgebrasCategory J D))) = - cone (λ v _ → 0a (snd (F-ob D v))) - (λ e → funExt (λ _ → F-hom D e .snd .pres0)) + cone (λ v _ → 0a (snd (F-ob D v))) + (λ e → funExt (λ _ → F-hom D e .snd .pres0)) 1a (snd (lim (LimitsCommAlgebrasCategory J D))) = - cone (λ v _ → 1a (snd (F-ob D v))) - (λ e → funExt (λ _ → F-hom D e .snd .pres1)) + cone (λ v _ → 1a (snd (F-ob D v))) + (λ e → funExt (λ _ → F-hom D e .snd .pres1)) _+_ (snd (lim (LimitsCommAlgebrasCategory J D))) x y = - cone (λ v _ → _+_ (snd (F-ob D v)) _ _) - ( λ e → funExt (λ _ → F-hom D e .snd .pres+ _ _ - ∙ λ i → _+_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) + cone (λ v _ → _+_ (snd (F-ob D v)) _ _) + ( λ e → funExt (λ _ → F-hom D e .snd .pres+ _ _ + ∙ λ i → _+_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) _·_ (snd (lim (LimitsCommAlgebrasCategory J D))) x y = - cone (λ v _ → _·_ (snd (F-ob D v)) _ _) - ( λ e → funExt (λ _ → F-hom D e .snd .pres· _ _ - ∙ λ i → _·_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) + cone (λ v _ → _·_ (snd (F-ob D v)) _ _) + ( λ e → funExt (λ _ → F-hom D e .snd .pres· _ _ + ∙ λ i → _·_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) (- snd (lim (LimitsCommAlgebrasCategory J D))) x = - cone (λ v _ → -_ (snd (F-ob D v)) _) - ( λ e → funExt (λ z → F-hom D e .snd .pres- _ - ∙ λ i → -_ (snd (F-ob D _)) (coneOutCommutes x e i tt*))) + cone (λ v _ → -_ (snd (F-ob D v)) _) + ( λ e → funExt (λ z → F-hom D e .snd .pres- _ + ∙ λ i → -_ (snd (F-ob D _)) (coneOutCommutes x e i tt*))) _⋆_ (snd (lim (LimitsCommAlgebrasCategory J D))) = λ r x → -- why can't Agda parse with r x on the lhs? - cone (λ v _ → _⋆_ (snd (F-ob D v)) r (coneOut x v tt*)) - ( λ e → funExt (λ z → F-hom D e .snd .pres⋆ _ _ - ∙ λ i → _⋆_ (snd (F-ob D _)) r (coneOutCommutes x e i tt*))) + cone (λ v _ → _⋆_ (snd (F-ob D v)) r (coneOut x v tt*)) + ( λ e → funExt (λ z → F-hom D e .snd .pres⋆ _ _ + ∙ λ i → _⋆_ (snd (F-ob D _)) r (coneOutCommutes x e i tt*))) isCommAlgebra (snd (lim (LimitsCommAlgebrasCategory J D))) = makeIsCommAlgebra - (isSetCone (ffA→SET ∘F D) (Unit* , _)) - (λ _ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Assoc _ _ _))) - (λ _ → cone≡ (λ v → funExt (λ _ → +IdR (snd (F-ob D v)) _))) - (λ _ → cone≡ (λ v → funExt (λ _ → +InvR (snd (F-ob D v)) _))) - (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Comm _ _))) - (λ _ _ _ → cone≡ λ v → funExt λ _ → ·Assoc (snd (F-ob D v)) _ _ _) - (λ _ → cone≡ (λ v → funExt (λ _ → ·IdL (snd (F-ob D v)) _))) - (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ·DistL+ (snd (F-ob D v)) _ _ _))) - (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .·Comm _ _))) - (λ _ _ _ → cone≡ λ v → funExt λ _ → ⋆Assoc (snd (F-ob D v)) _ _ _) - (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ⋆DistR+ (snd (F-ob D v)) _ _ _))) - (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ⋆DistL+ (snd (F-ob D v)) _ _ _))) - (λ _ → cone≡ (λ v → funExt (λ _ → ⋆IdL (snd (F-ob D v)) _))) - λ _ _ _ → cone≡ λ v → funExt λ _ → ⋆AssocL (snd (F-ob D v)) _ _ _ + (isSetCone (ffA→SET ∘F D) (Unit* , _)) + (λ _ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Assoc _ _ _))) + (λ _ → cone≡ (λ v → funExt (λ _ → +IdR (snd (F-ob D v)) _))) + (λ _ → cone≡ (λ v → funExt (λ _ → +InvR (snd (F-ob D v)) _))) + (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Comm _ _))) + (λ _ _ _ → cone≡ λ v → funExt λ _ → ·Assoc (snd (F-ob D v)) _ _ _) + (λ _ → cone≡ (λ v → funExt (λ _ → ·IdL (snd (F-ob D v)) _))) + (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ·DistL+ (snd (F-ob D v)) _ _ _))) + (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .·Comm _ _))) + (λ _ _ _ → cone≡ λ v → funExt λ _ → ⋆Assoc (snd (F-ob D v)) _ _ _) + (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ⋆DistR+ (snd (F-ob D v)) _ _ _))) + (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ⋆DistL+ (snd (F-ob D v)) _ _ _))) + (λ _ → cone≡ (λ v → funExt (λ _ → ⋆IdL (snd (F-ob D v)) _))) + λ _ _ _ → cone≡ λ v → funExt λ _ → ⋆AssocL (snd (F-ob D v)) _ _ _ fst (coneOut (limCone (LimitsCommAlgebrasCategory J D)) v) = - coneOut (limCone (completeSET J (funcComp ffA→SET D))) v + coneOut (limCone (completeSET J (funcComp ffA→SET D))) v pres0 (snd (coneOut (limCone (LimitsCommAlgebrasCategory J D)) v)) = refl pres1 (snd (coneOut (limCone (LimitsCommAlgebrasCategory J D)) v)) = refl pres+ (snd (coneOut (limCone (LimitsCommAlgebrasCategory J D)) v)) = λ _ _ → refl @@ -135,9 +135,9 @@ pres- (snd (coneOut (limCone (LimitsCommAlgebrasCategory J D)) v)) = λ _ → refl pres⋆ (snd (coneOut (limCone (LimitsCommAlgebrasCategory J D)) v)) = λ _ _ → refl coneOutCommutes (limCone (LimitsCommAlgebrasCategory J D)) e = - AlgebraHom≡ (coneOutCommutes (limCone (completeSET J (funcComp ffA→SET D))) e) + AlgebraHom≡ (coneOutCommutes (limCone (completeSET J (funcComp ffA→SET D))) e) univProp (LimitsCommAlgebrasCategory J D) c cc = uniqueExists - ( ( λ x → limArrow (completeSET J (funcComp ffA→SET D)) + ( ( λ x → limArrow (completeSET J (funcComp ffA→SET D)) (fst c , snd c .is-set) (cone (λ v _ → coneOut cc v .fst x) (λ e → funExt (λ _ → funExt⁻ (cong fst (coneOutCommutes cc e)) x))) x) @@ -148,7 +148,7 @@ (λ x y → cone≡ (λ v → funExt (λ _ → coneOut cc v .snd .pres⋆ _ _)))) (λ _ → AlgebraHom≡ refl) (isPropIsConeMor cc (limCone (LimitsCommAlgebrasCategory J D))) - (λ a' x → Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) + (λ a' x → Σ≡Prop (λ _ → isPropIsAlgebraHom _ _ _ _) (funExt (λ y → cone≡ λ v → funExt (λ _ → sym (funExt⁻ (cong fst (x v)) y))))) -- forgetful functor preserves limits @@ -162,10 +162,10 @@ where open isIso canonicalIso : (J : Category (ℓ-max ℓ ℓJ) ℓJ') - (D : Functor J CommAlgebrasCategory) + (D : Functor J CommAlgebrasCategory) → CatIso CommRingsCategory - (LimitsCommRingsCategory J (ffA→R ∘F D) .lim) - (ffA→R {ℓ' = theℓ} .F-ob (LimitsCommAlgebrasCategory J D .lim)) + (LimitsCommRingsCategory J (ffA→R ∘F D) .lim) + (ffA→R {ℓ' = theℓ} .F-ob (LimitsCommAlgebrasCategory J D .lim)) coneOut (fst (fst (canonicalIso J D)) cc) = coneOut cc coneOutCommutes (fst (fst (canonicalIso J D)) cc) = coneOutCommutes cc snd (fst (canonicalIso J D)) = makeIsRingHom refl (λ _ _ → refl) λ _ _ → refl @@ -175,7 +175,7 @@ sec (snd (canonicalIso J D)) = RingHom≡ refl ret (snd (canonicalIso J D)) = RingHom≡ refl - isConeMorCanonicalIso : ∀ J D → isConeMor (LimitsCommRingsCategory J (funcComp ffA→R D) .limCone) + isConeMorCanonicalIso : ∀ J D → isConeMor (LimitsCommRingsCategory J (funcComp ffA→R D) .limCone) (F-cone ffA→R (LimitsCommAlgebrasCategory J D .limCone)) (canonicalIso J D .fst) isConeMorCanonicalIso J D v = RingHom≡ refl @@ -231,8 +231,8 @@ univPropCommRingWithHomHom isRHom₁ isRHom₂ isRHom₃ isRHom₄ (E , f₅) (h₂ , comm₂) (h₁ , comm₁) squareComm = ((h₃ , h₃∘f₅≡f₁) , h₂≡g₂∘h₃ , h₁≡g₁∘h₃) - , λ h₃' → Σ≡Prop (λ _ → isProp× (isSetRingHom _ _ _ _) (isSetRingHom _ _ _ _)) - (Σ≡Prop (λ _ → isSetRingHom _ _ _ _) + , λ h₃' → Σ≡Prop (λ _ → isProp× (isSetRingHom _ _ _ _) (isSetRingHom _ _ _ _)) + (Σ≡Prop (λ _ → isSetRingHom _ _ _ _) (cong fst (commRingPB .univProp h₂ h₁ squareComm .snd (h₃' .fst .fst , h₃' .snd .fst , h₃' .snd .snd)))) where @@ -363,7 +363,7 @@ (h₂' ≡ toCommAlgebraHom _ _ g₂ isRHom₂ ∘a k') × (h₁' ≡ toCommAlgebraHom _ _ g₁ isRHom₁ ∘a k')) → (k , kComm₂ , kComm₁) ≡ y - uniqueness (k' , k'Comm₂ , k'Comm₁) = Σ≡Prop (λ _ → isProp× (isSetAlgebraHom _ _ _ _) + uniqueness (k' , k'Comm₂ , k'Comm₁) = Σ≡Prop (λ _ → isProp× (isSetAlgebraHom _ _ _ _) (isSetAlgebraHom _ _ _ _)) (AlgebraHom≡ (cong (fst ∘ fst ∘ fst) uniqHelper)) where @@ -398,23 +398,23 @@ module LimitFromCommRing {ℓJ ℓJ' : Level} (R A : CommRing ℓ) (J : Category ℓJ ℓJ') - (crDiag : Functor J (CommRingsCategory {ℓ = ℓ})) + (crDiag : Functor J (CommRingsCategory {ℓ = ℓ})) (crCone : Cone crDiag A) (toAlgCone : Cone crDiag R) where - open Functor + open Functor open Cone open AlgebraHoms open CommAlgChar R open CommAlgebraStr ⦃...⦄ - algDiag : Functor J (CommAlgebrasCategory R {ℓ' = ℓ}) - F-ob algDiag v = toCommAlg (F-ob crDiag v , coneOut toAlgCone v) - F-hom algDiag f = toCommAlgebraHom _ _ (F-hom crDiag f) (coneOutCommutes toAlgCone f) - F-id algDiag = AlgebraHom≡ (cong fst (F-id crDiag)) - F-seq algDiag f g = AlgebraHom≡ (cong fst (F-seq crDiag f g)) + algDiag : Functor J (CommAlgebrasCategory R {ℓ' = ℓ}) + F-ob algDiag v = toCommAlg (F-ob crDiag v , coneOut toAlgCone v) + F-hom algDiag f = toCommAlgebraHom _ _ (F-hom crDiag f) (coneOutCommutes toAlgCone f) + F-id algDiag = AlgebraHom≡ (cong fst (F-id crDiag)) + F-seq algDiag f g = AlgebraHom≡ (cong fst (F-seq crDiag f g)) module _ (φ : CommRingHom R A) (isConeMorφ : isConeMor toAlgCone crCone φ) where @@ -435,7 +435,7 @@ (χ , triangle) (univProp C cc .fst .snd) (λ _ → isPropIsConeMor _ _ _) - λ _ x → Σ≡Prop (λ _ → isSetRingHom _ _ _ _) + λ _ x → Σ≡Prop (λ _ → isSetRingHom _ _ _ _) (cong fst (univProp C cc .snd (_ , x))) where χ = univProp C cc .fst .fst @@ -473,7 +473,7 @@ where instance _ = snd D - _ = snd (F-ob algDiag v) + _ = snd (F-ob algDiag v) uniqueχ : ∃![ χ ∈ CommRingWithHomHom (C , ψ) (A , φ) ] isConeMor cc crCone (fst χ) uniqueχ = univPropWithHom _ cc isConeMorψ @@ -538,7 +538,7 @@ private ∥P∥ : ℙ (ob C) ∥P∥ x = ∥ P x ∥₁ , isPropPropTrunc - ΣC∥P∥Cat = ΣPropCat C ∥P∥ + ΣC∥P∥Cat = ΣPropCat C ∥P∥ CommAlgCat = CommAlgebrasCategory {ℓ = ℓ''} R {ℓ' = ℓ''} 𝓕UniqueEquiv : (x : ob C) (p q : P x) → isContr (CommAlgebraEquiv (𝓕 (x , p)) (𝓕 (x , q))) @@ -552,12 +552,12 @@ → (p : ∥ P x ∥₁) (q : ∥ P y ∥₁) → isContr (CommAlgebraHom (theMap y q) (theMap x p)) theAction _ _ f = elim2 (λ _ _ → isPropIsContr) λ _ _ → uniqueHom _ _ f - open Functor - universalPShf : Functor (ΣC∥P∥Cat ^op) CommAlgCat - F-ob universalPShf = uncurry theMap - F-hom universalPShf {x = x} {y = y} f = theAction _ _ f (y .snd) (x. snd) .fst - F-id universalPShf {x = x} = theAction (x .fst) (x .fst) (id C) (x .snd) (x .snd) .snd _ - F-seq universalPShf {x = x} {z = z} f g = theAction _ _ (g ⋆⟨ C ⟩ f) (z .snd) (x .snd) .snd _ + open Functor + universalPShf : Functor (ΣC∥P∥Cat ^op) CommAlgCat + F-ob universalPShf = uncurry theMap + F-hom universalPShf {x = x} {y = y} f = theAction _ _ f (y .snd) (x. snd) .fst + F-id universalPShf {x = x} = theAction (x .fst) (x .fst) (id C) (x .snd) (x .snd) .snd _ + F-seq universalPShf {x = x} {z = z} f g = theAction _ _ (g ⋆⟨ C ⟩ f) (z .snd) (x .snd) .snd _ -- a big transport to help verifying the pullback sheaf property @@ -573,26 +573,26 @@ -} (AlgCospan : Cospan CommAlgCat) (AlgPB : Pullback _ AlgCospan) - (p₁ : AlgPB .pbOb ≡ F-ob universalPShf x) (p₂ : AlgCospan .l ≡ F-ob universalPShf u) - (p₃ : AlgCospan .r ≡ F-ob universalPShf y) (p₄ : AlgCospan .m ≡ F-ob universalPShf v) + (p₁ : AlgPB .pbOb ≡ F-ob universalPShf x) (p₂ : AlgCospan .l ≡ F-ob universalPShf u) + (p₃ : AlgCospan .r ≡ F-ob universalPShf y) (p₄ : AlgCospan .m ≡ F-ob universalPShf v) where private -- just: 𝓕 k ⋆ 𝓕 f ≡ 𝓕 h ⋆ 𝓕 g - inducedSquare : seq' CommAlgCat {x = F-ob universalPShf x} - {y = F-ob universalPShf u} - {z = F-ob universalPShf v} - (F-hom universalPShf h) (F-hom universalPShf g) - ≡ seq' CommAlgCat {x = F-ob universalPShf x} - {y = F-ob universalPShf y} - {z = F-ob universalPShf v} - (F-hom universalPShf k) (F-hom universalPShf f) - inducedSquare = F-square universalPShf Csquare - - f' = F-hom universalPShf {x = y} {y = v} f - g' = F-hom universalPShf {x = u} {y = v} g - h' = F-hom universalPShf {x = x} {y = u} h - k' = F-hom universalPShf {x = x} {y = y} k + inducedSquare : seq' CommAlgCat {x = F-ob universalPShf x} + {y = F-ob universalPShf u} + {z = F-ob universalPShf v} + (F-hom universalPShf h) (F-hom universalPShf g) + ≡ seq' CommAlgCat {x = F-ob universalPShf x} + {y = F-ob universalPShf y} + {z = F-ob universalPShf v} + (F-hom universalPShf k) (F-hom universalPShf f) + inducedSquare = F-square universalPShf Csquare + + f' = F-hom universalPShf {x = y} {y = v} f + g' = F-hom universalPShf {x = u} {y = v} g + h' = F-hom universalPShf {x = x} {y = u} h + k' = F-hom universalPShf {x = x} {y = y} k gPathP : PathP (λ i → CommAlgCat [ p₂ i , p₄ i ]) (AlgCospan .s₁) g' gPathP = toPathP (sym (theAction _ _ g (v .snd) (u .snd) .snd _)) @@ -607,9 +607,9 @@ hPathP = toPathP (sym (theAction _ _ h (u .snd) (x .snd) .snd _)) fgCospan : Cospan CommAlgCat - l fgCospan = F-ob universalPShf u - m fgCospan = F-ob universalPShf v - r fgCospan = F-ob universalPShf y + l fgCospan = F-ob universalPShf u + m fgCospan = F-ob universalPShf v + r fgCospan = F-ob universalPShf y s₁ fgCospan = g' s₂ fgCospan = f' @@ -625,7 +625,7 @@ squarePathP = toPathP (CommAlgCat .isSetHom _ _ _ _) abstract - lemma : isPullback CommAlgCat fgCospan {c = F-ob universalPShf x} h' k' inducedSquare + lemma : isPullback CommAlgCat fgCospan {c = F-ob universalPShf x} h' k' inducedSquare lemma = transport (λ i → isPullback CommAlgCat (cospanPath i) {c = p₁ i} (hPathP i) (kPathP i) (squarePathP i)) (AlgPB .univProp) @@ -633,29 +633,29 @@ module toSheaf {J : Category ℓ'' ℓ''} - {D : Functor J (ΣC∥P∥Cat ^op)} {c : ob ΣC∥P∥Cat} (cc : Cone D c) -- will be B⋁Cone - {algDiag : Functor J CommAlgCat} - (algCone : Cone algDiag (F-ob universalPShf c)) - (p : (v : ob J) → F-ob algDiag v ≡ F-ob (universalPShf ∘F D) v) where + {D : Functor J (ΣC∥P∥Cat ^op)} {c : ob ΣC∥P∥Cat} (cc : Cone D c) -- will be B⋁Cone + {algDiag : Functor J CommAlgCat} + (algCone : Cone algDiag (F-ob universalPShf c)) + (p : (v : ob J) → F-ob algDiag v ≡ F-ob (universalPShf ∘F D) v) where open Cone private diagHomPathPs : ∀ {u v : ob J} (f : J [ u , v ]) → PathP (λ i → CommAlgebraHom (p u i) (p v i)) - (F-hom algDiag f) - (F-hom universalPShf (F-hom D f)) - diagHomPathPs f = toPathP (sym (theAction _ _ (F-hom D f) _ _ .snd _)) + (F-hom algDiag f) + (F-hom universalPShf (F-hom D f)) + diagHomPathPs f = toPathP (sym (theAction _ _ (F-hom D f) _ _ .snd _)) - diagPathAlg : algDiag ≡ universalPShf ∘F D - diagPathAlg = Functor≡ p diagHomPathPs + diagPathAlg : algDiag ≡ universalPShf ∘F D + diagPathAlg = Functor≡ p diagHomPathPs coneHomPathPs : ∀ (v : ob J) - → PathP (λ i → CommAlgebraHom (universalPShf .F-ob c) (diagPathAlg i .F-ob v)) + → PathP (λ i → CommAlgebraHom (universalPShf .F-ob c) (diagPathAlg i .F-ob v)) (algCone .coneOut v) (F-cone universalPShf cc .coneOut v) coneHomPathPs v = toPathP (sym (theAction _ _ (cc .coneOut v) _ _ .snd _)) - conePathPAlg : PathP (λ i → Cone (diagPathAlg i) (F-ob universalPShf c)) + conePathPAlg : PathP (λ i → Cone (diagPathAlg i) (F-ob universalPShf c)) algCone (F-cone universalPShf cc) conePathPAlg = conePathPDiag coneHomPathPs @@ -665,7 +665,7 @@ -- now for composition with forgetful functor CommRingsCat = CommRingsCategory {ℓ = ℓ''} Forgetful = ForgetfulCommAlgebra→CommRing {ℓ = ℓ''} R {ℓ' = ℓ''} - 𝓖 = Forgetful ∘F universalPShf + 𝓖 = Forgetful ∘F universalPShf module _ (isLimAlgCone : isLimCone _ _ algCone) where @@ -673,13 +673,13 @@ presLimForgetful : preservesLimits Forgetful presLimForgetful = ForgetfulCommAlgebra→CommRingPresLim R {ℓJ = ℓ''} {ℓJ' = ℓ''} - assocDiagPath : Forgetful ∘F (universalPShf ∘F D) ≡ 𝓖 ∘F D + assocDiagPath : Forgetful ∘F (universalPShf ∘F D) ≡ 𝓖 ∘F D assocDiagPath = F-assoc - conePathPCR : PathP (λ i → Cone (assocDiagPath i) (F-ob (Forgetful ∘F universalPShf) c)) + conePathPCR : PathP (λ i → Cone (assocDiagPath i) (F-ob (Forgetful ∘F universalPShf) c)) (F-cone Forgetful (F-cone universalPShf cc)) (F-cone 𝓖 cc) conePathPCR = conePathPDiag -- why does everything have to be explicit? - (λ v _ → (Forgetful ∘F universalPShf) .F-hom {x = c} {y = D .F-ob v} (cc .coneOut v)) + (λ v _ → (Forgetful ∘F universalPShf) .F-hom {x = c} {y = D .F-ob v} (cc .coneOut v)) toLimCone : isLimCone _ _ (F-cone 𝓖 cc) diff --git a/Cubical.Categories.Instances.CommRings.html b/Cubical.Categories.Instances.CommRings.html index a031203b01..a24af10fe4 100644 --- a/Cubical.Categories.Instances.CommRings.html +++ b/Cubical.Categories.Instances.CommRings.html @@ -31,7 +31,7 @@ open Category hiding (_∘_) open isUnivalent open isIso -open Functor +open Functor open CommRingStr open RingHoms open IsRingHom @@ -50,11 +50,11 @@ ⋆Assoc CommRingsCategory {R} {S} {T} {U} = compAssocCommRingHom {R = R} {S} {T} {U} isSetHom CommRingsCategory = isSetRingHom _ _ -ForgetfulCommRing→Set : Functor CommRingsCategory (SET ℓ) -F-ob ForgetfulCommRing→Set R = R .fst , CommRingStr.is-set (snd R) -F-hom ForgetfulCommRing→Set f x = f .fst x -F-id ForgetfulCommRing→Set = funExt (λ _ → refl) -F-seq ForgetfulCommRing→Set f g = funExt (λ _ → refl) +ForgetfulCommRing→Set : Functor CommRingsCategory (SET ℓ) +F-ob ForgetfulCommRing→Set R = R .fst , CommRingStr.is-set (snd R) +F-hom ForgetfulCommRing→Set f x = f .fst x +F-id ForgetfulCommRing→Set = funExt (λ _ → refl) +F-seq ForgetfulCommRing→Set f g = funExt (λ _ → refl) open Iso @@ -80,7 +80,7 @@ snd (inv CommRingIsoIsoCatIso e) = e .fst .snd rightInv CommRingIsoIsoCatIso x = CatIso≡ _ _ (RingHom≡ refl) leftInv (CommRingIsoIsoCatIso {R = R} {S}) x = - Σ≡Prop (λ x → isPropIsRingHom (CommRingStr→RingStr (R .snd)) + Σ≡Prop (λ x → isPropIsRingHom (CommRingStr→RingStr (R .snd)) (x .fun) (CommRingStr→RingStr (S .snd))) (Iso≡Set (is-set (snd R)) (is-set (snd S)) _ _ (λ _ → refl) (λ _ → refl)) @@ -133,48 +133,48 @@ LimitsCommRingsCategory : Limits {ℓJ} {ℓJ'} (CommRingsCategory {ℓ = ℓ-max ℓJ ℓJ'}) fst (lim (LimitsCommRingsCategory J D)) = - lim {J = J} (completeSET J (funcComp ForgetfulCommRing→Set D)) .fst + lim {J = J} (completeSET J (funcComp ForgetfulCommRing→Set D)) .fst 0r (snd (lim (LimitsCommRingsCategory J D))) = - cone (λ v _ → 0r (snd (F-ob D v))) - (λ e → funExt (λ _ → F-hom D e .snd .pres0)) + cone (λ v _ → 0r (snd (F-ob D v))) + (λ e → funExt (λ _ → F-hom D e .snd .pres0)) 1r (snd (lim (LimitsCommRingsCategory J D))) = - cone (λ v _ → 1r (snd (F-ob D v))) - (λ e → funExt (λ _ → F-hom D e .snd .pres1)) + cone (λ v _ → 1r (snd (F-ob D v))) + (λ e → funExt (λ _ → F-hom D e .snd .pres1)) _+_ (snd (lim (LimitsCommRingsCategory J D))) x y = - cone (λ v _ → _+_ (snd (F-ob D v)) _ _) - ( λ e → funExt (λ _ → F-hom D e .snd .pres+ _ _ - ∙ λ i → _+_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) + cone (λ v _ → _+_ (snd (F-ob D v)) _ _) + ( λ e → funExt (λ _ → F-hom D e .snd .pres+ _ _ + ∙ λ i → _+_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) _·_ (snd (lim (LimitsCommRingsCategory J D))) x y = - cone (λ v _ → _·_ (snd (F-ob D v)) _ _) - ( λ e → funExt (λ _ → F-hom D e .snd .pres· _ _ - ∙ λ i → _·_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) + cone (λ v _ → _·_ (snd (F-ob D v)) _ _) + ( λ e → funExt (λ _ → F-hom D e .snd .pres· _ _ + ∙ λ i → _·_ (snd (F-ob D _)) (coneOutCommutes x e i tt*) (coneOutCommutes y e i tt*))) (- snd (lim (LimitsCommRingsCategory J D))) x = - cone (λ v _ → -_ (snd (F-ob D v)) _) - ( λ e → funExt (λ z → F-hom D e .snd .pres- _ - ∙ λ i → -_ (snd (F-ob D _)) (coneOutCommutes x e i tt*))) + cone (λ v _ → -_ (snd (F-ob D v)) _) + ( λ e → funExt (λ z → F-hom D e .snd .pres- _ + ∙ λ i → -_ (snd (F-ob D _)) (coneOutCommutes x e i tt*))) isCommRing (snd (lim (LimitsCommRingsCategory J D))) = makeIsCommRing - (isSetCone (funcComp ForgetfulCommRing→Set D) (Unit* , _)) - (λ _ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Assoc _ _ _))) - (λ _ → cone≡ (λ v → funExt (λ _ → +IdR (snd (F-ob D v)) _))) - (λ _ → cone≡ (λ v → funExt (λ _ → +InvR (snd (F-ob D v)) _))) - (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Comm _ _))) - (λ _ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .·Assoc _ _ _))) - (λ _ → cone≡ (λ v → funExt (λ _ → ·IdR (snd (F-ob D v)) _))) - (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ·DistR+ (snd (F-ob D v)) _ _ _))) - (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .·Comm _ _))) + (isSetCone (funcComp ForgetfulCommRing→Set D) (Unit* , _)) + (λ _ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Assoc _ _ _))) + (λ _ → cone≡ (λ v → funExt (λ _ → +IdR (snd (F-ob D v)) _))) + (λ _ → cone≡ (λ v → funExt (λ _ → +InvR (snd (F-ob D v)) _))) + (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .+Comm _ _))) + (λ _ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .·Assoc _ _ _))) + (λ _ → cone≡ (λ v → funExt (λ _ → ·IdR (snd (F-ob D v)) _))) + (λ _ _ _ → cone≡ (λ v → funExt (λ _ → ·DistR+ (snd (F-ob D v)) _ _ _))) + (λ _ _ → cone≡ (λ v → funExt (λ _ → snd (F-ob D v) .·Comm _ _))) fst (coneOut (limCone (LimitsCommRingsCategory J D)) v) = - coneOut (limCone (completeSET J (funcComp ForgetfulCommRing→Set D))) v + coneOut (limCone (completeSET J (funcComp ForgetfulCommRing→Set D))) v pres0 (snd (coneOut (limCone (LimitsCommRingsCategory J D)) v)) = refl pres1 (snd (coneOut (limCone (LimitsCommRingsCategory J D)) v)) = refl pres+ (snd (coneOut (limCone (LimitsCommRingsCategory J D)) v)) = λ _ _ → refl pres· (snd (coneOut (limCone (LimitsCommRingsCategory J D)) v)) = λ _ _ → refl pres- (snd (coneOut (limCone (LimitsCommRingsCategory J D)) v)) = λ _ → refl coneOutCommutes (limCone (LimitsCommRingsCategory J D)) e = - RingHom≡ (coneOutCommutes (limCone (completeSET J (funcComp ForgetfulCommRing→Set D))) e) + RingHom≡ (coneOutCommutes (limCone (completeSET J (funcComp ForgetfulCommRing→Set D))) e) univProp (LimitsCommRingsCategory J D) c cc = uniqueExists - ( (λ x → limArrow (completeSET J (funcComp ForgetfulCommRing→Set D)) + ( (λ x → limArrow (completeSET J (funcComp ForgetfulCommRing→Set D)) (fst c , snd c .is-set) (cone (λ v _ → coneOut cc v .fst x) (λ e → funExt (λ _ → funExt⁻ (cong fst (coneOutCommutes cc e)) x))) x) @@ -184,6 +184,6 @@ (λ x y → cone≡ (λ v → funExt (λ _ → coneOut cc v .snd .pres· _ _)))) (λ _ → RingHom≡ refl) (isPropIsConeMor cc (limCone (LimitsCommRingsCategory J D))) - (λ a' x → Σ≡Prop (λ _ → isPropIsRingHom _ _ _) + (λ a' x → Σ≡Prop (λ _ → isPropIsRingHom _ _ _) (funExt (λ y → cone≡ λ v → funExt (λ _ → sym (funExt⁻ (cong fst (x v)) y))))) \ No newline at end of file diff --git a/Cubical.Categories.Instances.Discrete.html b/Cubical.Categories.Instances.Discrete.html index d595781e20..bcdc09c37f 100644 --- a/Cubical.Categories.Instances.Discrete.html +++ b/Cubical.Categories.Instances.Discrete.html @@ -32,16 +32,16 @@ module _ {A : hGroupoid ℓ} {C : Category ℓC ℓC'} where - open Functor + open Functor -- Functions f: A → ob C give functors F: DiscreteCategory A → C - DiscFunc : (fst A → ob C) → Functor (DiscreteCategory A) C - DiscFunc f .F-ob = f - DiscFunc f .F-hom {x} p = subst (λ z → C [ f x , f z ]) p (id C) - DiscFunc f .F-id {x} = substRefl {B = λ z → C [ f x , f z ]} (id C) + DiscFunc : (fst A → ob C) → Functor (DiscreteCategory A) C + DiscFunc f .F-ob = f + DiscFunc f .F-hom {x} p = subst (λ z → C [ f x , f z ]) p (id C) + DiscFunc f .F-id {x} = substRefl {B = λ z → C [ f x , f z ]} (id C) -- Preserves composition - DiscFunc f .F-seq {x} {y} p q = + DiscFunc f .F-seq {x} {y} p q = let open Category C using () renaming (_⋆_ to _●_) in let Hom[fx,f—] = (λ (w : A .fst) → C [ f x , f w ]) in diff --git a/Cubical.Categories.Instances.EilenbergMoore.html b/Cubical.Categories.Instances.EilenbergMoore.html index 552ad42333..424cda0cca 100644 --- a/Cubical.Categories.Instances.EilenbergMoore.html +++ b/Cubical.Categories.Instances.EilenbergMoore.html @@ -9,7 +9,7 @@ open import Cubical.Foundations.Univalence open import Cubical.Categories.Category -open import Cubical.Categories.Functor renaming (𝟙⟨_⟩ to funcId) +open import Cubical.Categories.Functor renaming (𝟙⟨_⟩ to funcId) open import Cubical.Categories.NaturalTransformation.Base open import Cubical.Categories.Monad.Base open import Cubical.Categories.Instances.FunctorAlgebras @@ -23,13 +23,13 @@ module _ {C : Category ℓC ℓC'} (monadM : Monad C) where private - M : Functor C C + M : Functor C C M = fst monadM --open Category private module C = Category C - open Functor + open Functor open NatTrans open IsMonad (snd monadM) @@ -39,7 +39,7 @@ open Algebra algA field str-η : str C.∘ N-ob η carrier ≡ C.id - str-μ : str C.∘ N-ob μ carrier ≡ str C.∘ F-hom M str + str-μ : str C.∘ N-ob μ carrier ≡ str C.∘ F-hom M str open IsEMAlgebra @@ -55,66 +55,66 @@ -- cannot simplify level: --lossy-unification won't allow it. EMCategory = FullSubcategory (AlgebrasCategory M) IsEMAlgebra - ForgetEM : Functor EMCategory (AlgebrasCategory M) + ForgetEM : Functor EMCategory (AlgebrasCategory M) ForgetEM = FullInclusion (AlgebrasCategory M) IsEMAlgebra - ForgetEMAlgebra : Functor EMCategory C - ForgetEMAlgebra = funcComp (ForgetAlgebra M) ForgetEM + ForgetEMAlgebra : Functor EMCategory C + ForgetEMAlgebra = funcComp (ForgetAlgebra M) ForgetEM open Algebra freeEMAlgebra : C.ob → EMAlgebra - carrier (fst (freeEMAlgebra x)) = F-ob M x + carrier (fst (freeEMAlgebra x)) = F-ob M x str (fst (freeEMAlgebra x)) = N-ob μ x str-η (snd (freeEMAlgebra x)) = lemma - where lemma : N-ob η (F-ob M x) C.⋆ N-ob μ x ≡ C.id + where lemma : N-ob η (F-ob M x) C.⋆ N-ob μ x ≡ C.id lemma = funExt⁻ (congP (λ i → N-ob) idl-μ) x str-μ (snd (freeEMAlgebra x)) = lemma - where lemma : N-ob μ (F-ob M x) C.⋆ N-ob μ x ≡ F-hom M (N-ob μ x) C.⋆ N-ob μ x + where lemma : N-ob μ (F-ob M x) C.⋆ N-ob μ x ≡ F-hom M (N-ob μ x) C.⋆ N-ob μ x lemma = funExt⁻ (congP (λ i → N-ob) (symP-fromGoal assoc-μ)) x open AlgebraHom - FreeEMAlgebra : Functor C EMCategory - F-ob FreeEMAlgebra x = freeEMAlgebra x - carrierHom (F-hom FreeEMAlgebra {x} {y} φ) = F-hom M φ - strHom (F-hom FreeEMAlgebra {x} {y} φ) = sym (N-hom μ φ) - F-id FreeEMAlgebra = AlgebraHom≡ M (F-id M) - F-seq FreeEMAlgebra {x} {y} {z} φ ψ = AlgebraHom≡ M (F-seq M φ ψ) + FreeEMAlgebra : Functor C EMCategory + F-ob FreeEMAlgebra x = freeEMAlgebra x + carrierHom (F-hom FreeEMAlgebra {x} {y} φ) = F-hom M φ + strHom (F-hom FreeEMAlgebra {x} {y} φ) = sym (N-hom μ φ) + F-id FreeEMAlgebra = AlgebraHom≡ M (F-id M) + F-seq FreeEMAlgebra {x} {y} {z} φ ψ = AlgebraHom≡ M (F-seq M φ ψ) - ForgetFreeEMAlgebra : funcComp ForgetEMAlgebra FreeEMAlgebra ≡ M - ForgetFreeEMAlgebra = Functor≡ (λ x → refl) (λ f → refl) + ForgetFreeEMAlgebra : funcComp ForgetEMAlgebra FreeEMAlgebra ≡ M + ForgetFreeEMAlgebra = Functor≡ (λ x → refl) (λ f → refl) - emCounit : NatTrans (funcComp FreeEMAlgebra ForgetEMAlgebra) (funcId EMCategory) + emCounit : NatTrans (funcComp FreeEMAlgebra ForgetEMAlgebra) (funcId EMCategory) carrierHom (N-ob emCounit (algebra A α , isEMA)) = α strHom (N-ob emCounit (algebra A α , isEMA)) = str-μ isEMA N-hom emCounit {algebra A α , isEMA} {algebra B β , isEMB} (algebraHom f isalgF) = AlgebraHom≡ M (sym (isalgF)) - open NaturalBijection - open _⊣_ + open NaturalBijection + open _⊣_ open _≅_ emBijection : ∀ a emB → - (EMCategory [ FreeEMAlgebra ⟅ a ⟆ , emB ]) ≅ (C [ a , ForgetEMAlgebra ⟅ emB ⟆ ]) + (EMCategory [ FreeEMAlgebra ⟅ a ⟆ , emB ]) ≅ (C [ a , ForgetEMAlgebra ⟅ emB ⟆ ]) fun (emBijection a (algebra b β , isEMB)) (algebraHom f isalgF) = f C.∘ N-ob η a - carrierHom (inv (emBijection a (algebra b β , isEMB)) f) = β C.∘ F-hom M f + carrierHom (inv (emBijection a (algebra b β , isEMB)) f) = β C.∘ F-hom M f strHom (inv (emBijection a (algebra b β , isEMB)) f) = - (N-ob μ a C.⋆ (F-hom M f C.⋆ β)) + (N-ob μ a C.⋆ (F-hom M f C.⋆ β)) ≡⟨ sym (C.⋆Assoc _ _ _) ⟩ - ((N-ob μ a C.⋆ F-hom M f) C.⋆ β) + ((N-ob μ a C.⋆ F-hom M f) C.⋆ β) ≡⟨ cong (C._⋆ β) (sym (N-hom μ f)) ⟩ - ((F-hom M (F-hom M f) C.⋆ N-ob μ b) C.⋆ β) + ((F-hom M (F-hom M f) C.⋆ N-ob μ b) C.⋆ β) ≡⟨ C.⋆Assoc _ _ _ ⟩ - (F-hom M (F-hom M f) C.⋆ (N-ob μ b C.⋆ β)) - ≡⟨ cong (F-hom M (F-hom M f) C.⋆_) (str-μ isEMB) ⟩ - (F-hom M (F-hom M f) C.⋆ (F-hom M β C.⋆ β)) + (F-hom M (F-hom M f) C.⋆ (N-ob μ b C.⋆ β)) + ≡⟨ cong (F-hom M (F-hom M f) C.⋆_) (str-μ isEMB) ⟩ + (F-hom M (F-hom M f) C.⋆ (F-hom M β C.⋆ β)) ≡⟨ sym (C.⋆Assoc _ _ _) ⟩ - ((F-hom M (F-hom M f) C.⋆ F-hom M β) C.⋆ β) - ≡⟨ cong (C._⋆ β) (sym (F-seq M _ _)) ⟩ - (F-hom M (F-hom M f C.⋆ β) C.⋆ β) ∎ + ((F-hom M (F-hom M f) C.⋆ F-hom M β) C.⋆ β) + ≡⟨ cong (C._⋆ β) (sym (F-seq M _ _)) ⟩ + (F-hom M (F-hom M f C.⋆ β) C.⋆ β) ∎ rightInv (emBijection a (algebra b β , isEMB)) f = - (N-ob η a C.⋆ (F-hom M f C.⋆ β)) + (N-ob η a C.⋆ (F-hom M f C.⋆ β)) ≡⟨ sym (C.⋆Assoc _ _ _) ⟩ - ((N-ob η a C.⋆ F-hom M f) C.⋆ β) + ((N-ob η a C.⋆ F-hom M f) C.⋆ β) ≡⟨ cong (C._⋆ β) (sym (N-hom η f)) ⟩ ((f C.⋆ N-ob η b) C.⋆ β) ≡⟨ C.⋆Assoc _ _ _ ⟩ @@ -124,43 +124,43 @@ ≡⟨ C.⋆IdR _ ⟩ f ∎ leftInv (emBijection a (algebra b β , isEMB)) (algebraHom f isalgF) = AlgebraHom≡ M ( - (F-hom M (N-ob η a C.⋆ f) C.⋆ β) - ≡⟨ cong (C._⋆ β) (F-seq M _ _) ⟩ - ((F-hom M (N-ob η a) C.⋆ F-hom M f) C.⋆ β) + (F-hom M (N-ob η a C.⋆ f) C.⋆ β) + ≡⟨ cong (C._⋆ β) (F-seq M _ _) ⟩ + ((F-hom M (N-ob η a) C.⋆ F-hom M f) C.⋆ β) ≡⟨ C.⋆Assoc _ _ _ ⟩ - (F-hom M (N-ob η a) C.⋆ (F-hom M f C.⋆ β)) - ≡⟨ cong (F-hom M (N-ob η a) C.⋆_) (sym isalgF) ⟩ - (F-hom M (N-ob η a) C.⋆ (N-ob μ a C.⋆ f)) + (F-hom M (N-ob η a) C.⋆ (F-hom M f C.⋆ β)) + ≡⟨ cong (F-hom M (N-ob η a) C.⋆_) (sym isalgF) ⟩ + (F-hom M (N-ob η a) C.⋆ (N-ob μ a C.⋆ f)) ≡⟨ sym (C.⋆Assoc _ _ _) ⟩ - ((F-hom M (N-ob η a) C.⋆ N-ob μ a) C.⋆ f) + ((F-hom M (N-ob η a) C.⋆ N-ob μ a) C.⋆ f) ≡⟨ cong (C._⋆ f) (funExt⁻ (congP (λ i → N-ob) idr-μ) a) ⟩ (C.id C.⋆ f) ≡⟨ C.⋆IdL f ⟩ f ∎ ) - emAdjunction : FreeEMAlgebra ⊣ ForgetEMAlgebra - adjIso emAdjunction {a} {algebra b β , isEMB} = emBijection a (algebra b β , isEMB) - adjNatInD emAdjunction {a} {algebra b β , isEMB} {algebra c γ , isEMC} + emAdjunction : FreeEMAlgebra ⊣ ForgetEMAlgebra + adjIso emAdjunction {a} {algebra b β , isEMB} = emBijection a (algebra b β , isEMB) + adjNatInD emAdjunction {a} {algebra b β , isEMB} {algebra c γ , isEMC} (algebraHom f isalgF) (algebraHom g isalgG) = sym (C.⋆Assoc _ _ _) - adjNatInC emAdjunction {a} {b} {algebra c γ , isEMC} f g = AlgebraHom≡ M ( - (F-hom M (g C.⋆ f) C.⋆ γ) - ≡⟨ cong (C._⋆ γ) (F-seq M _ _) ⟩ - ((F-hom M g C.⋆ F-hom M f) C.⋆ γ) + adjNatInC emAdjunction {a} {b} {algebra c γ , isEMC} f g = AlgebraHom≡ M ( + (F-hom M (g C.⋆ f) C.⋆ γ) + ≡⟨ cong (C._⋆ γ) (F-seq M _ _) ⟩ + ((F-hom M g C.⋆ F-hom M f) C.⋆ γ) ≡⟨ C.⋆Assoc _ _ _ ⟩ - (F-hom M g C.⋆ (F-hom M f C.⋆ γ)) ∎ + (F-hom M g C.⋆ (F-hom M f C.⋆ γ)) ∎ ) module _ {C : Category ℓC ℓC'} {monadM monadN : Monad C} (monadν : MonadHom monadM monadN) where open Category C - open Functor + open Functor open IsEMAlgebra open NatTrans private - M N : Functor C C + M N : Functor C C M = fst monadM N = fst monadN module M = IsMonad (snd monadM) @@ -169,7 +169,7 @@ ν = fst monadν module ν = IsMonadHom (snd monadν) - mapIsEMAlgebra : (algA : Algebra N) → IsEMAlgebra monadN algA → IsEMAlgebra monadM (F-ob (AlgebrasFunctor ν) algA) + mapIsEMAlgebra : (algA : Algebra N) → IsEMAlgebra monadN algA → IsEMAlgebra monadM (F-ob (AlgebrasFunctor ν) algA) str-η (mapIsEMAlgebra (algebra a αN) isEMA) = N-ob M.η a ⋆ (N-ob ν a ⋆ αN) ≡⟨ sym (⋆Assoc _ _ _) ⟩ @@ -183,25 +183,25 @@ ≡⟨ sym (⋆Assoc _ _ _) ⟩ (N-ob M.μ a ⋆ N-ob ν a) ⋆ αN ≡⟨ cong (_⋆ αN) (cong (λ θ → N-ob θ a) ν.N-μ) ⟩ - ((F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ N-ob N.μ a) ⋆ αN + ((F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ N-ob N.μ a) ⋆ αN ≡⟨ ⋆Assoc _ _ _ ⟩ - (F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ (N-ob N.μ a ⋆ αN) - ≡⟨ cong ((F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆_) (isEMA .str-μ) ⟩ - (F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ (F-hom N αN ⋆ αN) + (F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ (N-ob N.μ a ⋆ αN) + ≡⟨ cong ((F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆_) (isEMA .str-μ) ⟩ + (F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ (F-hom N αN ⋆ αN) ≡⟨ sym (⋆Assoc _ _ _) ⟩ - ((F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ F-hom N αN) ⋆ αN + ((F-hom M (N-ob ν a) ⋆ N-ob ν (F-ob N a)) ⋆ F-hom N αN) ⋆ αN ≡⟨ cong (_⋆ αN) (⋆Assoc _ _ _) ⟩ - (F-hom M (N-ob ν a) ⋆ (N-ob ν (F-ob N a) ⋆ F-hom N αN)) ⋆ αN - ≡⟨ cong (_⋆ αN) (cong (F-hom M (N-ob ν a) ⋆_) (sym (N-hom ν αN))) ⟩ - (F-hom M (N-ob ν a) ⋆ (F-hom M αN ⋆ N-ob ν a)) ⋆ αN + (F-hom M (N-ob ν a) ⋆ (N-ob ν (F-ob N a) ⋆ F-hom N αN)) ⋆ αN + ≡⟨ cong (_⋆ αN) (cong (F-hom M (N-ob ν a) ⋆_) (sym (N-hom ν αN))) ⟩ + (F-hom M (N-ob ν a) ⋆ (F-hom M αN ⋆ N-ob ν a)) ⋆ αN ≡⟨ cong (_⋆ αN) (sym (⋆Assoc _ _ _)) ⟩ - ((F-hom M (N-ob ν a) ⋆ F-hom M αN) ⋆ N-ob ν a) ⋆ αN - ≡⟨ cong (_⋆ αN) (cong (_⋆ N-ob ν a) (sym (F-seq M _ _))) ⟩ - (F-hom M (N-ob ν a ⋆ αN) ⋆ N-ob ν a) ⋆ αN + ((F-hom M (N-ob ν a) ⋆ F-hom M αN) ⋆ N-ob ν a) ⋆ αN + ≡⟨ cong (_⋆ αN) (cong (_⋆ N-ob ν a) (sym (F-seq M _ _))) ⟩ + (F-hom M (N-ob ν a ⋆ αN) ⋆ N-ob ν a) ⋆ αN ≡⟨ ⋆Assoc _ _ _ ⟩ - F-hom M (N-ob ν a ⋆ αN) ⋆ (N-ob ν a ⋆ αN) ∎ + F-hom M (N-ob ν a ⋆ αN) ⋆ (N-ob ν a ⋆ αN) ∎ - EMFunctor : Functor (EMCategory monadN) (EMCategory monadM) + EMFunctor : Functor (EMCategory monadN) (EMCategory monadM) EMFunctor = MapFullSubcategory (AlgebrasCategory N) (IsEMAlgebra monadN) (AlgebrasCategory M) (IsEMAlgebra monadM) diff --git a/Cubical.Categories.Instances.FunctorAlgebras.html b/Cubical.Categories.Instances.FunctorAlgebras.html index cbe611213f..41be1f8637 100644 --- a/Cubical.Categories.Instances.FunctorAlgebras.html +++ b/Cubical.Categories.Instances.FunctorAlgebras.html @@ -17,13 +17,13 @@ private variable ℓC ℓC' : Level -module _ {C : Category ℓC ℓC'} (F : Functor C C) where +module _ {C : Category ℓC ℓC'} (F : Functor C C) where open Category - open Functor + open Functor IsAlgebra : ob C → Type ℓC' - IsAlgebra x = C [ F-ob F x , x ] + IsAlgebra x = C [ F-ob F x , x ] record Algebra : Type (ℓ-max ℓC ℓC') where constructor algebra @@ -33,7 +33,7 @@ open Algebra IsAlgebraHom : (algA algB : Algebra) → C [ carrier algA , carrier algB ] → Type ℓC' - IsAlgebraHom algA algB f = (f ∘⟨ C ⟩ str algA) ≡ (str algB ∘⟨ C ⟩ F-hom F f) + IsAlgebraHom algA algB f = (f ∘⟨ C ⟩ str algA) ≡ (str algB ∘⟨ C ⟩ F-hom F f) record AlgebraHom (algA algB : Algebra) : Type ℓC' where constructor algebraHom @@ -58,7 +58,7 @@ → (carrierHom algF ≡ carrierHom algG) → algF ≡ algG carrierHom (AlgebraHom≡ {algA} {algB} {algF} {algG} p i) = p i strHom (AlgebraHom≡ {algA} {algB} {algF} {algG} p i) = idfun - (PathP (λ j → (p j ∘⟨ C ⟩ str algA) ≡ (str algB ∘⟨ C ⟩ F-hom F (p j))) + (PathP (λ j → (p j ∘⟨ C ⟩ str algA) ≡ (str algB ∘⟨ C ⟩ F-hom F (p j))) (strHom algF) (strHom algG) ) @@ -70,7 +70,7 @@ idAlgebraHom : {algA : Algebra} → AlgebraHom algA algA carrierHom (idAlgebraHom {algA}) = id C strHom (idAlgebraHom {algA}) = - ⋆IdR C (str algA) ∙∙ sym (⋆IdL C (str algA)) ∙∙ cong (λ φ → φ ⋆⟨ C ⟩ str algA) (sym (F-id F)) + ⋆IdR C (str algA) ∙∙ sym (⋆IdL C (str algA)) ∙∙ cong (λ φ → φ ⋆⟨ C ⟩ str algA) (sym (F-id F)) seqAlgebraHom : {algA algB algC : Algebra} (algF : AlgebraHom algA algB) (algG : AlgebraHom algB algC) @@ -81,15 +81,15 @@ ≡⟨ sym (⋆Assoc C (str algA) (carrierHom algF) (carrierHom algG)) ⟩ (str algA ⋆⟨ C ⟩ carrierHom algF) ⋆⟨ C ⟩ carrierHom algG ≡⟨ cong (λ φ → φ ⋆⟨ C ⟩ carrierHom algG) (strHom algF) ⟩ - (F-hom F (carrierHom algF) ⋆⟨ C ⟩ str algB) ⋆⟨ C ⟩ carrierHom algG - ≡⟨ ⋆Assoc C (F-hom F (carrierHom algF)) (str algB) (carrierHom algG) ⟩ - F-hom F (carrierHom algF) ⋆⟨ C ⟩ (str algB ⋆⟨ C ⟩ carrierHom algG) - ≡⟨ cong (λ φ → F-hom F (carrierHom algF) ⋆⟨ C ⟩ φ) (strHom algG) ⟩ - F-hom F (carrierHom algF) ⋆⟨ C ⟩ (F-hom F (carrierHom algG) ⋆⟨ C ⟩ str algC) - ≡⟨ sym (⋆Assoc C (F-hom F (carrierHom algF)) (F-hom F (carrierHom algG)) (str algC)) ⟩ - (F-hom F (carrierHom algF) ⋆⟨ C ⟩ F-hom F (carrierHom algG)) ⋆⟨ C ⟩ str algC - ≡⟨ cong (λ φ → φ ⋆⟨ C ⟩ str algC) (sym (F-seq F (carrierHom algF) (carrierHom algG))) ⟩ - F-hom F (carrierHom algF ⋆⟨ C ⟩ carrierHom algG) ⋆⟨ C ⟩ str algC ∎ + (F-hom F (carrierHom algF) ⋆⟨ C ⟩ str algB) ⋆⟨ C ⟩ carrierHom algG + ≡⟨ ⋆Assoc C (F-hom F (carrierHom algF)) (str algB) (carrierHom algG) ⟩ + F-hom F (carrierHom algF) ⋆⟨ C ⟩ (str algB ⋆⟨ C ⟩ carrierHom algG) + ≡⟨ cong (λ φ → F-hom F (carrierHom algF) ⋆⟨ C ⟩ φ) (strHom algG) ⟩ + F-hom F (carrierHom algF) ⋆⟨ C ⟩ (F-hom F (carrierHom algG) ⋆⟨ C ⟩ str algC) + ≡⟨ sym (⋆Assoc C (F-hom F (carrierHom algF)) (F-hom F (carrierHom algG)) (str algC)) ⟩ + (F-hom F (carrierHom algF) ⋆⟨ C ⟩ F-hom F (carrierHom algG)) ⋆⟨ C ⟩ str algC + ≡⟨ cong (λ φ → φ ⋆⟨ C ⟩ str algC) (sym (F-seq F (carrierHom algF) (carrierHom algG))) ⟩ + F-hom F (carrierHom algF ⋆⟨ C ⟩ carrierHom algG) ⋆⟨ C ⟩ str algC ∎ AlgebrasCategory : Category (ℓ-max ℓC ℓC') ℓC' ob AlgebrasCategory = Algebra @@ -103,35 +103,35 @@ isSetHom AlgebrasCategory = subst isSet (sym (pathRepAlgebraHom _ _)) (isSetΣ (isSetHom C) (λ f → isProp→isSet (isSetHom C _ _))) - ForgetAlgebra : Functor AlgebrasCategory C - F-ob ForgetAlgebra = carrier - F-hom ForgetAlgebra = carrierHom - F-id ForgetAlgebra = refl - F-seq ForgetAlgebra algF algG = refl + ForgetAlgebra : Functor AlgebrasCategory C + F-ob ForgetAlgebra = carrier + F-hom ForgetAlgebra = carrierHom + F-id ForgetAlgebra = refl + F-seq ForgetAlgebra algF algG = refl -module _ {C : Category ℓC ℓC'} {F G : Functor C C} (τ : NatTrans F G) where +module _ {C : Category ℓC ℓC'} {F G : Functor C C} (τ : NatTrans F G) where private module C = Category C - open Functor + open Functor open NatTrans open AlgebraHom - AlgebrasFunctor : Functor (AlgebrasCategory G) (AlgebrasCategory F) - F-ob AlgebrasFunctor (algebra a α) = algebra a (α C.∘ N-ob τ a) - carrierHom (F-hom AlgebrasFunctor (algebraHom f isalgF)) = f - strHom (F-hom AlgebrasFunctor {algebra a α} {algebra b β} (algebraHom f isalgF)) = + AlgebrasFunctor : Functor (AlgebrasCategory G) (AlgebrasCategory F) + F-ob AlgebrasFunctor (algebra a α) = algebra a (α C.∘ N-ob τ a) + carrierHom (F-hom AlgebrasFunctor (algebraHom f isalgF)) = f + strHom (F-hom AlgebrasFunctor {algebra a α} {algebra b β} (algebraHom f isalgF)) = (N-ob τ a C.⋆ α) C.⋆ f ≡⟨ C.⋆Assoc (N-ob τ a) α f ⟩ N-ob τ a C.⋆ (α C.⋆ f) ≡⟨ cong (N-ob τ a C.⋆_) isalgF ⟩ - N-ob τ a C.⋆ (F-hom G f C.⋆ β) + N-ob τ a C.⋆ (F-hom G f C.⋆ β) ≡⟨ sym (C.⋆Assoc _ _ _) ⟩ - (N-ob τ a C.⋆ F-hom G f) C.⋆ β + (N-ob τ a C.⋆ F-hom G f) C.⋆ β ≡⟨ cong (C._⋆ β) (sym (N-hom τ f)) ⟩ - (F-hom F f C.⋆ N-ob τ b) C.⋆ β + (F-hom F f C.⋆ N-ob τ b) C.⋆ β ≡⟨ C.⋆Assoc _ _ _ ⟩ - F-hom F f C.⋆ (N-ob τ b C.⋆ β) ∎ - F-id AlgebrasFunctor = AlgebraHom≡ F refl - F-seq AlgebrasFunctor algΦ algΨ = AlgebraHom≡ F refl + F-hom F f C.⋆ (N-ob τ b C.⋆ β) ∎ + F-id AlgebrasFunctor = AlgebraHom≡ F refl + F-seq AlgebrasFunctor algΦ algΨ = AlgebraHom≡ F refl \ No newline at end of file diff --git a/Cubical.Categories.Instances.Functors.Endo.html b/Cubical.Categories.Instances.Functors.Endo.html index c2aad67819..728392aa77 100644 --- a/Cubical.Categories.Instances.Functors.Endo.html +++ b/Cubical.Categories.Instances.Functors.Endo.html @@ -5,7 +5,7 @@ open import Cubical.Categories.Category.Base open import Cubical.Categories.Monoidal.Base open import Cubical.Categories.Instances.Functors -open import Cubical.Categories.Functor.Base renaming (𝟙⟨_⟩ to idfunctor) +open import Cubical.Categories.Functor.Base renaming (𝟙⟨_⟩ to idfunctor) open import Cubical.Categories.NaturalTransformation.Base open import Cubical.Categories.NaturalTransformation.Properties @@ -13,7 +13,7 @@ open Category open NatTrans - open Functor + open Functor open StrictMonStr open TensorStr diff --git a/Cubical.Categories.Instances.Functors.html b/Cubical.Categories.Instances.Functors.html index bc15fd8330..95fefdd2cb 100644 --- a/Cubical.Categories.Instances.Functors.html +++ b/Cubical.Categories.Instances.Functors.html @@ -34,31 +34,31 @@ module _ (C : Category ℓC ℓC') (D : Category ℓD ℓD') where open Category open NatTrans - open Functor + open Functor FUNCTOR : Category (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) (ℓ-max (ℓ-max ℓC ℓC') ℓD') - ob FUNCTOR = Functor C D + ob FUNCTOR = Functor C D Hom[_,_] FUNCTOR = NatTrans id FUNCTOR {F} = idTrans F _⋆_ FUNCTOR = seqTrans ⋆IdL FUNCTOR α = makeNatTransPath λ i x → D .⋆IdL (α .N-ob x) i ⋆IdR FUNCTOR α = makeNatTransPath λ i x → D .⋆IdR (α .N-ob x) i ⋆Assoc FUNCTOR α β γ = makeNatTransPath λ i x → D .⋆Assoc (α .N-ob x) (β .N-ob x) (γ .N-ob x) i - isSetHom FUNCTOR = isSetNatTrans + isSetHom FUNCTOR = isSetNatTrans open isIsoC renaming (inv to invC) -- componentwise iso is an iso in Functor - FUNCTORIso : ∀ {F G : Functor C D} (α : F ⇒ G) + FUNCTORIso : ∀ {F G : Functor C D} (α : F ⇒ G) → (∀ (c : C .ob) → isIsoC D (α ⟦ c ⟧)) → isIsoC FUNCTOR α FUNCTORIso α is .invC .N-ob c = (is c) .invC FUNCTORIso {F} {G} α is .invC .N-hom {c} {d} f = invMoveL areInv-αc - ( α ⟦ c ⟧ ⋆⟨ D ⟩ (G ⟪ f ⟫ ⋆⟨ D ⟩ is d .invC) + ( α ⟦ c ⟧ ⋆⟨ D ⟩ (G ⟪ f ⟫ ⋆⟨ D ⟩ is d .invC) ≡⟨ sym (D .⋆Assoc _ _ _) ⟩ - (α ⟦ c ⟧ ⋆⟨ D ⟩ G ⟪ f ⟫) ⋆⟨ D ⟩ is d .invC + (α ⟦ c ⟧ ⋆⟨ D ⟩ G ⟪ f ⟫) ⋆⟨ D ⟩ is d .invC ≡⟨ sym (invMoveR areInv-αd (α .N-hom f)) ⟩ - F ⟪ f ⟫ + F ⟪ f ⟫ ∎ ) where areInv-αc : areInv _ (α ⟦ c ⟧) ((is c) .invC) @@ -70,7 +70,7 @@ FUNCTORIso α is .ret = makeNatTransPath (funExt (λ c → (is c) .ret)) -- iso is componentwise iso in Functor - FUNCTORIso' : ∀ {F G : Functor C D} (α : F ⇒ G) + FUNCTORIso' : ∀ {F G : Functor C D} (α : F ⇒ G) → isIsoC FUNCTOR α → ((c : C .ob) → isIsoC D (α ⟦ c ⟧)) FUNCTORIso' α isom c .invC = isom .invC .N-ob c @@ -80,21 +80,21 @@ open Iso open NatIso - FUNCTORIso→NatIso : {F G : Functor C D} → CatIso FUNCTOR F G → NatIso F G + FUNCTORIso→NatIso : {F G : Functor C D} → CatIso FUNCTOR F G → NatIso F G FUNCTORIso→NatIso α .trans = α .fst FUNCTORIso→NatIso α .nIso = FUNCTORIso' _ (α .snd) - NatIso→FUNCTORIso : {F G : Functor C D} → NatIso F G → CatIso FUNCTOR F G + NatIso→FUNCTORIso : {F G : Functor C D} → NatIso F G → CatIso FUNCTOR F G NatIso→FUNCTORIso α = α .trans , FUNCTORIso _ (α .nIso) - Path→FUNCTORIso→NatIso : {F G : Functor C D} → (p : F ≡ G) → pathToNatIso p ≡ FUNCTORIso→NatIso (pathToIso p) + Path→FUNCTORIso→NatIso : {F G : Functor C D} → (p : F ≡ G) → pathToNatIso p ≡ FUNCTORIso→NatIso (pathToIso p) Path→FUNCTORIso→NatIso {F = F} p = J (λ _ p → pathToNatIso p ≡ FUNCTORIso→NatIso (pathToIso p)) (NatIso≡ refl-helper) p where refl-helper : _ - refl-helper i x = ((λ i → pathToIso-refl {C = D} {x = F .F-ob x} i .fst) + refl-helper i x = ((λ i → pathToIso-refl {C = D} {x = F .F-ob x} i .fst) ∙ (λ i → pathToIso-refl {C = FUNCTOR} {x = F} (~ i) .fst .N-ob x)) i - Iso-FUNCTORIso-NatIso : {F G : Functor C D} → Iso (CatIso FUNCTOR F G) (NatIso F G) + Iso-FUNCTORIso-NatIso : {F G : Functor C D} → Iso (CatIso FUNCTOR F G) (NatIso F G) Iso-FUNCTORIso-NatIso .fun = FUNCTORIso→NatIso Iso-FUNCTORIso-NatIso .inv = NatIso→FUNCTORIso Iso-FUNCTORIso-NatIso .rightInv α i .trans = α .trans @@ -104,7 +104,7 @@ Iso-FUNCTORIso-NatIso .leftInv α i .snd = isProp→PathP (λ i → isPropIsIso _) (FUNCTORIso _ (FUNCTORIso' _ (α .snd))) (α .snd) i - FUNCTORIso≃NatIso : {F G : Functor C D} → CatIso FUNCTOR F G ≃ NatIso F G + FUNCTORIso≃NatIso : {F G : Functor C D} → CatIso FUNCTOR F G ≃ NatIso F G FUNCTORIso≃NatIso = isoToEquiv Iso-FUNCTORIso-NatIso @@ -115,59 +115,59 @@ isUnivalentFUNCTOR : isUnivalent D → isUnivalent FUNCTOR isUnivalentFUNCTOR isUnivD .univ _ _ = isEquiv[equivFunA≃B∘f]→isEquiv[f] _ FUNCTORIso≃NatIso - (subst isEquiv (λ i p → Path→FUNCTORIso→NatIso p i) (Path≃NatIso isUnivD .snd)) - - appF : Functor (FUNCTOR ×C C) D - appF .F-ob (F , c) = F ⟅ c ⟆ - appF .F-hom {F , c} {G , d} (α , f) = α .N-ob d ∘⟨ D ⟩ F .F-hom f - appF .F-id {F , c} = - D .id ∘⟨ D ⟩ F .F-hom (C .id) ≡⟨ D .⋆IdR (F .F-hom (C .id)) ⟩ - F .F-hom (C .id) ≡⟨ F .F-id ⟩ + (subst isEquiv (λ i p → Path→FUNCTORIso→NatIso p i) (Path≃NatIso isUnivD .snd)) + + appF : Functor (FUNCTOR ×C C) D + appF .F-ob (F , c) = F ⟅ c ⟆ + appF .F-hom {F , c} {G , d} (α , f) = α .N-ob d ∘⟨ D ⟩ F .F-hom f + appF .F-id {F , c} = + D .id ∘⟨ D ⟩ F .F-hom (C .id) ≡⟨ D .⋆IdR (F .F-hom (C .id)) ⟩ + F .F-hom (C .id) ≡⟨ F .F-id ⟩ D .id ∎ - appF .F-seq {F , c}{G , d}{H , e} (α , f) (β , g ) = - (β .N-ob e ∘⟨ D ⟩ α .N-ob e) ∘⟨ D ⟩ F .F-hom (g ∘⟨ C ⟩ f) - ≡⟨ (λ i → (β .N-ob e ∘⟨ D ⟩ α .N-ob e) ∘⟨ D ⟩ F .F-seq f g i) ⟩ - (β .N-ob e ∘⟨ D ⟩ α .N-ob e) ∘⟨ D ⟩ (F .F-hom g ∘⟨ D ⟩ F .F-hom f) + appF .F-seq {F , c}{G , d}{H , e} (α , f) (β , g ) = + (β .N-ob e ∘⟨ D ⟩ α .N-ob e) ∘⟨ D ⟩ F .F-hom (g ∘⟨ C ⟩ f) + ≡⟨ (λ i → (β .N-ob e ∘⟨ D ⟩ α .N-ob e) ∘⟨ D ⟩ F .F-seq f g i) ⟩ + (β .N-ob e ∘⟨ D ⟩ α .N-ob e) ∘⟨ D ⟩ (F .F-hom g ∘⟨ D ⟩ F .F-hom f) ≡⟨ sym (D .⋆Assoc _ _ _) ⟩ - β .N-ob e ∘⟨ D ⟩ (α .N-ob e ∘⟨ D ⟩ (F .F-hom g ∘⟨ D ⟩ F .F-hom f)) + β .N-ob e ∘⟨ D ⟩ (α .N-ob e ∘⟨ D ⟩ (F .F-hom g ∘⟨ D ⟩ F .F-hom f)) ≡⟨ (λ i → β .N-ob e - ∘⟨ D ⟩ D .⋆Assoc (F .F-hom f) (F .F-hom g) (α .N-ob e) i) ⟩ - β .N-ob e ∘⟨ D ⟩ ((α .N-ob e ∘⟨ D ⟩ F .F-hom g) ∘⟨ D ⟩ F .F-hom f) - ≡⟨ (λ i → β .N-ob e ∘⟨ D ⟩ α .N-hom g i ∘⟨ D ⟩ F .F-hom f) ⟩ - β .N-ob e ∘⟨ D ⟩ ((G .F-hom g ∘⟨ D ⟩ α .N-ob d) ∘⟨ D ⟩ F .F-hom f) + ∘⟨ D ⟩ D .⋆Assoc (F .F-hom f) (F .F-hom g) (α .N-ob e) i) ⟩ + β .N-ob e ∘⟨ D ⟩ ((α .N-ob e ∘⟨ D ⟩ F .F-hom g) ∘⟨ D ⟩ F .F-hom f) + ≡⟨ (λ i → β .N-ob e ∘⟨ D ⟩ α .N-hom g i ∘⟨ D ⟩ F .F-hom f) ⟩ + β .N-ob e ∘⟨ D ⟩ ((G .F-hom g ∘⟨ D ⟩ α .N-ob d) ∘⟨ D ⟩ F .F-hom f) ≡⟨ (λ i → β .N-ob e - ∘⟨ D ⟩ D .⋆Assoc (F .F-hom f) (α .N-ob d) (G .F-hom g) (~ i) ) ⟩ - β .N-ob e ∘⟨ D ⟩ (G .F-hom g ∘⟨ D ⟩ (α .N-ob d ∘⟨ D ⟩ F .F-hom f)) + ∘⟨ D ⟩ D .⋆Assoc (F .F-hom f) (α .N-ob d) (G .F-hom g) (~ i) ) ⟩ + β .N-ob e ∘⟨ D ⟩ (G .F-hom g ∘⟨ D ⟩ (α .N-ob d ∘⟨ D ⟩ F .F-hom f)) ≡⟨ D .⋆Assoc _ _ _ ⟩ - (β .N-ob e ∘⟨ D ⟩ G .F-hom g) ∘⟨ D ⟩ (α .N-ob d ∘⟨ D ⟩ F .F-hom f) ∎ + (β .N-ob e ∘⟨ D ⟩ G .F-hom g) ∘⟨ D ⟩ (α .N-ob d ∘⟨ D ⟩ F .F-hom f) ∎ module _ (E : Category ℓE ℓE') where - λF : Functor (E ×C C) D → Functor E FUNCTOR - λF F .F-ob e .F-ob c = F ⟅ e , c ⟆ - λF F .F-ob e .F-hom f = F ⟪ (E .id) , f ⟫ - λF F .F-ob e .F-id = F .F-id - λF F .F-ob e .F-seq f g = - F ⟪ E .id , g ∘⟨ C ⟩ f ⟫ - ≡⟨ (λ i → F ⟪ (E .⋆IdL (E .id) (~ i)) , (g ∘⟨ C ⟩ f) ⟫) ⟩ - (F ⟪ (E .id ∘⟨ E ⟩ E .id) , g ∘⟨ C ⟩ f ⟫) - ≡⟨ F .F-seq (E .id , f) (E .id , g) ⟩ - (F ⟪ E .id , g ⟫ ∘⟨ D ⟩ F ⟪ E .id , f ⟫) ∎ - λF F .F-hom h .N-ob c = F ⟪ h , (C .id) ⟫ - λF F .F-hom h .N-hom f = - F ⟪ h , C .id ⟫ ∘⟨ D ⟩ F ⟪ E .id , f ⟫ ≡⟨ sym (F .F-seq _ _) ⟩ - F ⟪ h ∘⟨ E ⟩ E .id , C .id ∘⟨ C ⟩ f ⟫ - ≡⟨ (λ i → F ⟪ E .⋆IdL h i , C .⋆IdR f i ⟫) ⟩ - F ⟪ h , f ⟫ ≡⟨ (λ i → F ⟪ (E .⋆IdR h (~ i)) , (C .⋆IdL f (~ i)) ⟫) ⟩ - F ⟪ E .id ∘⟨ E ⟩ h , f ∘⟨ C ⟩ C .id ⟫ ≡⟨ F .F-seq _ _ ⟩ - F ⟪ E .id , f ⟫ ∘⟨ D ⟩ F ⟪ h , C .id ⟫ ∎ - λF F .F-id = makeNatTransPath (funExt λ c → F .F-id) - λF F .F-seq f g = makeNatTransPath (funExt lem) where + λF : Functor (E ×C C) D → Functor E FUNCTOR + λF F .F-ob e .F-ob c = F ⟅ e , c ⟆ + λF F .F-ob e .F-hom f = F ⟪ (E .id) , f ⟫ + λF F .F-ob e .F-id = F .F-id + λF F .F-ob e .F-seq f g = + F ⟪ E .id , g ∘⟨ C ⟩ f ⟫ + ≡⟨ (λ i → F ⟪ (E .⋆IdL (E .id) (~ i)) , (g ∘⟨ C ⟩ f) ⟫) ⟩ + (F ⟪ (E .id ∘⟨ E ⟩ E .id) , g ∘⟨ C ⟩ f ⟫) + ≡⟨ F .F-seq (E .id , f) (E .id , g) ⟩ + (F ⟪ E .id , g ⟫ ∘⟨ D ⟩ F ⟪ E .id , f ⟫) ∎ + λF F .F-hom h .N-ob c = F ⟪ h , (C .id) ⟫ + λF F .F-hom h .N-hom f = + F ⟪ h , C .id ⟫ ∘⟨ D ⟩ F ⟪ E .id , f ⟫ ≡⟨ sym (F .F-seq _ _) ⟩ + F ⟪ h ∘⟨ E ⟩ E .id , C .id ∘⟨ C ⟩ f ⟫ + ≡⟨ (λ i → F ⟪ E .⋆IdL h i , C .⋆IdR f i ⟫) ⟩ + F ⟪ h , f ⟫ ≡⟨ (λ i → F ⟪ (E .⋆IdR h (~ i)) , (C .⋆IdL f (~ i)) ⟫) ⟩ + F ⟪ E .id ∘⟨ E ⟩ h , f ∘⟨ C ⟩ C .id ⟫ ≡⟨ F .F-seq _ _ ⟩ + F ⟪ E .id , f ⟫ ∘⟨ D ⟩ F ⟪ h , C .id ⟫ ∎ + λF F .F-id = makeNatTransPath (funExt λ c → F .F-id) + λF F .F-seq f g = makeNatTransPath (funExt lem) where lem : (c : C .ob) → - F ⟪ g ∘⟨ E ⟩ f , C .id ⟫ ≡ - F ⟪ g , C .id ⟫ ∘⟨ D ⟩ F ⟪ f , C .id ⟫ + F ⟪ g ∘⟨ E ⟩ f , C .id ⟫ ≡ + F ⟪ g , C .id ⟫ ∘⟨ D ⟩ F ⟪ f , C .id ⟫ lem c = - F ⟪ g ∘⟨ E ⟩ f , C .id ⟫ - ≡⟨ (λ i → F ⟪ (g ∘⟨ E ⟩ f) , (C .⋆IdR (C .id) (~ i)) ⟫) ⟩ - F ⟪ g ∘⟨ E ⟩ f , C .id ∘⟨ C ⟩ C .id ⟫ - ≡⟨ F .F-seq (f , C .id) (g , C .id) ⟩ - (F ⟪ g , C .id ⟫) ∘⟨ D ⟩ (F ⟪ f , C .id ⟫) ∎ + F ⟪ g ∘⟨ E ⟩ f , C .id ⟫ + ≡⟨ (λ i → F ⟪ (g ∘⟨ E ⟩ f) , (C .⋆IdR (C .id) (~ i)) ⟫) ⟩ + F ⟪ g ∘⟨ E ⟩ f , C .id ∘⟨ C ⟩ C .id ⟫ + ≡⟨ F .F-seq (f , C .id) (g , C .id) ⟩ + (F ⟪ g , C .id ⟫) ∘⟨ D ⟩ (F ⟪ f , C .id ⟫) ∎ \ No newline at end of file diff --git a/Cubical.Categories.Instances.Groups.html b/Cubical.Categories.Instances.Groups.html new file mode 100644 index 0000000000..db128e85df --- /dev/null +++ b/Cubical.Categories.Instances.Groups.html @@ -0,0 +1,77 @@ + +Cubical.Categories.Instances.Groups
-- The category Grp of groups
+{-# OPTIONS --safe #-}
+
+module Cubical.Categories.Instances.Groups where
+
+open import Cubical.Foundations.Prelude
+open import Cubical.Foundations.Equiv
+open import Cubical.Foundations.Isomorphism
+open import Cubical.Foundations.Function
+
+open import Cubical.Algebra.Group.Base
+open import Cubical.Algebra.Group.GroupPath
+open import Cubical.Algebra.Group.Morphisms
+open import Cubical.Algebra.Group.MorphismProperties
+
+open import Cubical.Categories.Category.Base renaming (isIso to isCatIso)
+open import Cubical.Categories.Instances.Sets
+open import Cubical.Categories.Functor.Base
+
+open import Cubical.Data.Sigma
+
+module _ {ℓ : Level} where
+  open Category hiding (_∘_)
+
+  GroupCategory : Category (ℓ-suc ℓ) ℓ
+  GroupCategory .ob = Group ℓ
+  GroupCategory .Hom[_,_] = GroupHom
+  GroupCategory .id = idGroupHom
+  GroupCategory ._⋆_ = compGroupHom
+  GroupCategory .⋆IdL f = GroupHom≡ refl
+  GroupCategory .⋆IdR f = GroupHom≡ refl
+  GroupCategory .⋆Assoc f g h = GroupHom≡ refl
+  GroupCategory .isSetHom = isSetGroupHom
+
+  Forget : Functor GroupCategory (SET ℓ)
+  Functor.F-ob Forget G = fst G , GroupStr.is-set (snd G)
+  Functor.F-hom Forget = fst
+  Functor.F-id Forget = refl
+  Functor.F-seq Forget _ _ = refl
+
+  GroupsCatIso≃GroupEquiv : ∀ G G' →
+    CatIso GroupCategory G G' ≃
+     GroupEquiv G G'
+  GroupsCatIso≃GroupEquiv G G' =
+      Σ-cong-equiv-snd
+        (λ _ → propBiimpl→Equiv
+          (isPropIsIso _) (isPropIsEquiv _)
+           (isoToIsEquiv ∘ →iso) →ciso)
+       ∙ₑ Σ-assoc-swap-≃
+   where
+    open Iso
+    open isCatIso
+    →iso : ∀ {x} → isCatIso GroupCategory x → Iso _ _
+    fun (→iso ici) = _
+    inv (→iso ici) = fst (inv ici)
+    rightInv (→iso ici) b i = fst (sec ici i) b
+    leftInv (→iso ici) a i = fst (ret ici i) a
+
+    →ciso : ∀ {x} → isEquiv (fst x) → isCatIso GroupCategory x
+    fst (inv (→ciso is≃)) = _
+    snd (inv (→ciso {x} is≃)) =
+      isGroupHomInv ((_ , is≃) , (snd x))
+    sec (→ciso is≃) =
+     Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (secEq (_ , is≃)))
+    ret (→ciso is≃) =
+     Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (retEq (_ , is≃)))
+
+
+  isUnivalentGrp : isUnivalent {ℓ' = ℓ} GroupCategory
+  isUnivalent.univ isUnivalentGrp _ _ =
+   precomposesToId→Equiv _ _ (funExt
+               (Σ≡Prop isPropIsIso
+              ∘ Σ≡Prop (λ _ → isPropIsGroupHom _ _)
+              ∘ λ _ → transportRefl _))
+              (snd (GroupsCatIso≃GroupEquiv _ _ ∙ₑ GroupPath _ _))
+
\ No newline at end of file diff --git a/Cubical.Categories.Instances.Sets.html b/Cubical.Categories.Instances.Sets.html index 1b1114fcb5..b997849235 100644 --- a/Cubical.Categories.Instances.Sets.html +++ b/Cubical.Categories.Instances.Sets.html @@ -33,29 +33,29 @@ variable ℓ ℓ' : Level -open Functor +open Functor -- Hom functors -_[-,_] : (C : Category ℓ ℓ') → (c : C .ob) → Functor (C ^op) (SET ℓ') -(C [-, c ]) .F-ob x = (C [ x , c ]) , C .isSetHom -(C [-, c ]) .F-hom f k = f ⋆⟨ C ⟩ k -(C [-, c ]) .F-id = funExt λ _ → C .⋆IdL _ -(C [-, c ]) .F-seq _ _ = funExt λ _ → C .⋆Assoc _ _ _ - -_[_,-] : (C : Category ℓ ℓ') → (c : C .ob)→ Functor C (SET ℓ') -(C [ c ,-]) .F-ob x = (C [ c , x ]) , C .isSetHom -(C [ c ,-]) .F-hom f k = k ⋆⟨ C ⟩ f -(C [ c ,-]) .F-id = funExt λ _ → C .⋆IdR _ -(C [ c ,-]) .F-seq _ _ = funExt λ _ → sym (C .⋆Assoc _ _ _) +_[-,_] : (C : Category ℓ ℓ') → (c : C .ob) → Functor (C ^op) (SET ℓ') +(C [-, c ]) .F-ob x = (C [ x , c ]) , C .isSetHom +(C [-, c ]) .F-hom f k = f ⋆⟨ C ⟩ k +(C [-, c ]) .F-id = funExt λ _ → C .⋆IdL _ +(C [-, c ]) .F-seq _ _ = funExt λ _ → C .⋆Assoc _ _ _ + +_[_,-] : (C : Category ℓ ℓ') → (c : C .ob)→ Functor C (SET ℓ') +(C [ c ,-]) .F-ob x = (C [ c , x ]) , C .isSetHom +(C [ c ,-]) .F-hom f k = k ⋆⟨ C ⟩ f +(C [ c ,-]) .F-id = funExt λ _ → C .⋆IdR _ +(C [ c ,-]) .F-seq _ _ = funExt λ _ → sym (C .⋆Assoc _ _ _) -- Lift functor -LiftF : Functor (SET ℓ) (SET (ℓ-max ℓ ℓ')) -LiftF {ℓ}{ℓ'} .F-ob A = (Lift {ℓ}{ℓ'} (A .fst)) , isOfHLevelLift 2 (A .snd) -LiftF .F-hom f x = lift (f (x .lower)) -LiftF .F-id = refl -LiftF .F-seq f g = funExt λ x → refl +LiftF : Functor (SET ℓ) (SET (ℓ-max ℓ ℓ')) +LiftF {ℓ}{ℓ'} .F-ob A = (Lift {ℓ}{ℓ'} (A .fst)) , isOfHLevelLift 2 (A .snd) +LiftF .F-hom f x = lift (f (x .lower)) +LiftF .F-id = refl +LiftF .F-seq f g = funExt λ x → refl -module _ {C : Category ℓ ℓ'} {F : Functor C (SET ℓ')} where +module _ {C : Category ℓ ℓ'} {F : Functor C (SET ℓ')} where open NatTrans -- natural transformations by pre/post composition @@ -69,7 +69,7 @@ ≡[ i ]⟨ (λ l → (α ⟦ d ⟧) (⋆Assoc C f l k (~ i))) ⟩ (λ l → (α ⟦ d ⟧) (f ⋆⟨ C ⟩ l ⋆⟨ C ⟩ k)) ≡[ i ]⟨ (λ l → (α .N-hom k) i (f ⋆⟨ C ⟩ l)) ⟩ - (λ l → (F ⟪ k ⟫) ((α ⟦ c ⟧) (f ⋆⟨ C ⟩ l))) + (λ l → (F ⟪ k ⟫) ((α ⟦ c ⟧) (f ⋆⟨ C ⟩ l))) ∎ -- properties diff --git a/Cubical.Categories.Instances.Terminal.html b/Cubical.Categories.Instances.Terminal.html index e66438d85c..6e8708d394 100644 --- a/Cubical.Categories.Instances.Terminal.html +++ b/Cubical.Categories.Instances.Terminal.html @@ -17,11 +17,11 @@ module _ {ℓC ℓ'C : Level} {C : Category ℓC ℓ'C} where private module C = Category C - open Functor + open Functor - FunctorFromTerminal : {ℓ* : Level} → C.ob → Functor (TerminalCategory {ℓ*}) C - FunctorFromTerminal c .F-ob _ = c - FunctorFromTerminal c .F-hom _ = C.id - FunctorFromTerminal c .F-id = refl - FunctorFromTerminal c .F-seq _ _ = sym $ C.⋆IdL C.id + FunctorFromTerminal : {ℓ* : Level} → C.ob → Functor (TerminalCategory {ℓ*}) C + FunctorFromTerminal c .F-ob _ = c + FunctorFromTerminal c .F-hom _ = C.id + FunctorFromTerminal c .F-id = refl + FunctorFromTerminal c .F-seq _ _ = sym $ C.⋆IdL C.id \ No newline at end of file diff --git a/Cubical.Categories.Instances.ZFunctors.html b/Cubical.Categories.Instances.ZFunctors.html index eff8dd8443..be7d772e2e 100644 --- a/Cubical.Categories.Instances.ZFunctors.html +++ b/Cubical.Categories.Instances.ZFunctors.html @@ -55,18 +55,18 @@ module _ {ℓ : Level} where - open Functor + open Functor open NatTrans open CommRingStr ⦃...⦄ open IsRingHom -- using the naming conventions of Nieper-Wisskirchen - ℤFunctor = Functor (CommRingsCategory {ℓ = ℓ}) (SET ℓ) + ℤFunctor = Functor (CommRingsCategory {ℓ = ℓ}) (SET ℓ) ℤFUNCTOR = FUNCTOR (CommRingsCategory {ℓ = ℓ}) (SET ℓ) -- Yoneda in the notation of Demazure-Gabriel, -- uses that double op is original category definitionally - Sp : Functor (CommRingsCategory {ℓ = ℓ} ^op) ℤFUNCTOR + Sp : Functor (CommRingsCategory {ℓ = ℓ} ^op) ℤFUNCTOR Sp = YO {C = (CommRingsCategory {ℓ = ℓ} ^op)} -- the forgetful functor @@ -76,62 +76,62 @@ 𝔸¹ = ForgetfulCommRing→Set -- the global sections functor - Γ : Functor ℤFUNCTOR (CommRingsCategory {ℓ = ℓ-suc ℓ} ^op) - fst (F-ob Γ X) = X ⇒ 𝔸¹ + Γ : Functor ℤFUNCTOR (CommRingsCategory {ℓ = ℓ-suc ℓ} ^op) + fst (F-ob Γ X) = X ⇒ 𝔸¹ -- ring struncture induced by internal ring object 𝔸¹ - N-ob (CommRingStr.0r (snd (F-ob Γ X))) A _ = 0r + N-ob (CommRingStr.0r (snd (F-ob Γ X))) A _ = 0r where instance _ = A .snd - N-hom (CommRingStr.0r (snd (F-ob Γ X))) φ = funExt λ _ → sym (φ .snd .pres0) + N-hom (CommRingStr.0r (snd (F-ob Γ X))) φ = funExt λ _ → sym (φ .snd .pres0) - N-ob (CommRingStr.1r (snd (F-ob Γ X))) A _ = 1r + N-ob (CommRingStr.1r (snd (F-ob Γ X))) A _ = 1r where instance _ = A .snd - N-hom (CommRingStr.1r (snd (F-ob Γ X))) φ = funExt λ _ → sym (φ .snd .pres1) + N-hom (CommRingStr.1r (snd (F-ob Γ X))) φ = funExt λ _ → sym (φ .snd .pres1) - N-ob ((snd (F-ob Γ X) CommRingStr.+ α) β) A x = α .N-ob A x + β .N-ob A x + N-ob ((snd (F-ob Γ X) CommRingStr.+ α) β) A x = α .N-ob A x + β .N-ob A x where instance _ = A .snd - N-hom ((snd (F-ob Γ X) CommRingStr.+ α) β) {x = A} {y = B} φ = funExt path + N-hom ((snd (F-ob Γ X) CommRingStr.+ α) β) {x = A} {y = B} φ = funExt path where instance _ = A .snd _ = B .snd - path : ∀ x → α .N-ob B (X .F-hom φ x) + β .N-ob B (X .F-hom φ x) + path : ∀ x → α .N-ob B (X .F-hom φ x) + β .N-ob B (X .F-hom φ x) ≡ φ .fst (α .N-ob A x + β .N-ob A x) - path x = α .N-ob B (X .F-hom φ x) + β .N-ob B (X .F-hom φ x) + path x = α .N-ob B (X .F-hom φ x) + β .N-ob B (X .F-hom φ x) ≡⟨ cong₂ _+_ (funExt⁻ (α .N-hom φ) x) (funExt⁻ (β .N-hom φ) x) ⟩ φ .fst (α .N-ob A x) + φ .fst (β .N-ob A x) ≡⟨ sym (φ .snd .pres+ _ _) ⟩ φ .fst (α .N-ob A x + β .N-ob A x) ∎ - N-ob ((snd (F-ob Γ X) CommRingStr.· α) β) A x = α .N-ob A x · β .N-ob A x + N-ob ((snd (F-ob Γ X) CommRingStr.· α) β) A x = α .N-ob A x · β .N-ob A x where instance _ = A .snd - N-hom ((snd (F-ob Γ X) CommRingStr.· α) β) {x = A} {y = B} φ = funExt path + N-hom ((snd (F-ob Γ X) CommRingStr.· α) β) {x = A} {y = B} φ = funExt path where instance _ = A .snd _ = B .snd - path : ∀ x → α .N-ob B (X .F-hom φ x) · β .N-ob B (X .F-hom φ x) + path : ∀ x → α .N-ob B (X .F-hom φ x) · β .N-ob B (X .F-hom φ x) ≡ φ .fst (α .N-ob A x · β .N-ob A x) - path x = α .N-ob B (X .F-hom φ x) · β .N-ob B (X .F-hom φ x) + path x = α .N-ob B (X .F-hom φ x) · β .N-ob B (X .F-hom φ x) ≡⟨ cong₂ _·_ (funExt⁻ (α .N-hom φ) x) (funExt⁻ (β .N-hom φ) x) ⟩ φ .fst (α .N-ob A x) · φ .fst (β .N-ob A x) ≡⟨ sym (φ .snd .pres· _ _) ⟩ φ .fst (α .N-ob A x · β .N-ob A x) ∎ - N-ob ((CommRingStr.- snd (F-ob Γ X)) α) A x = - α .N-ob A x + N-ob ((CommRingStr.- snd (F-ob Γ X)) α) A x = - α .N-ob A x where instance _ = A .snd - N-hom ((CommRingStr.- snd (F-ob Γ X)) α) {x = A} {y = B} φ = funExt path + N-hom ((CommRingStr.- snd (F-ob Γ X)) α) {x = A} {y = B} φ = funExt path where instance _ = A .snd _ = B .snd - path : ∀ x → - α .N-ob B (X .F-hom φ x) ≡ φ .fst (- α .N-ob A x) - path x = - α .N-ob B (X .F-hom φ x) ≡⟨ cong -_ (funExt⁻ (α .N-hom φ) x) ⟩ + path : ∀ x → - α .N-ob B (X .F-hom φ x) ≡ φ .fst (- α .N-ob A x) + path x = - α .N-ob B (X .F-hom φ x) ≡⟨ cong -_ (funExt⁻ (α .N-hom φ) x) ⟩ - φ .fst (α .N-ob A x) ≡⟨ sym (φ .snd .pres- _) ⟩ φ .fst (- α .N-ob A x) ∎ - CommRingStr.isCommRing (snd (F-ob Γ X)) = makeIsCommRing - isSetNatTrans + CommRingStr.isCommRing (snd (F-ob Γ X)) = makeIsCommRing + isSetNatTrans (λ _ _ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.+Assoc _ _ _)) (λ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.+IdR _)) (λ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.+InvR _)) @@ -142,28 +142,28 @@ (λ _ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.·Comm _ _)) -- action on natural transformations - fst (F-hom Γ α) = α ●ᵛ_ - pres0 (snd (F-hom Γ α)) = makeNatTransPath (funExt₂ λ _ _ → refl) - pres1 (snd (F-hom Γ α)) = makeNatTransPath (funExt₂ λ _ _ → refl) - pres+ (snd (F-hom Γ α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) - pres· (snd (F-hom Γ α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) - pres- (snd (F-hom Γ α)) _ = makeNatTransPath (funExt₂ λ _ _ → refl) + fst (F-hom Γ α) = α ●ᵛ_ + pres0 (snd (F-hom Γ α)) = makeNatTransPath (funExt₂ λ _ _ → refl) + pres1 (snd (F-hom Γ α)) = makeNatTransPath (funExt₂ λ _ _ → refl) + pres+ (snd (F-hom Γ α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) + pres· (snd (F-hom Γ α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) + pres- (snd (F-hom Γ α)) _ = makeNatTransPath (funExt₂ λ _ _ → refl) -- functoriality of Γ - F-id Γ = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) - F-seq Γ _ _ = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) + F-id Γ = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) + F-seq Γ _ _ = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) -- we get an adjunction Γ ⊣ Sp modulo size issues module AdjBij where - open Functor + open Functor open NatTrans open Iso open IsRingHom private module _ {A : CommRing ℓ} {X : ℤFunctor {ℓ}} where - _♭ : CommRingHom A (Γ .F-ob X) → X ⇒ Sp .F-ob A + _♭ : CommRingHom A (Γ .F-ob X) → X ⇒ Sp .F-ob A fst (N-ob (φ ♭) B x) a = φ .fst a .N-ob B x pres0 (snd (N-ob (φ ♭) B x)) = cong (λ y → y .N-ob B x) (φ .snd .pres0) @@ -176,7 +176,7 @@ -- the other direction is just precomposition modulo Yoneda - _♯ : X ⇒ Sp .F-ob A → CommRingHom A (Γ .F-ob X) + _♯ : X ⇒ Sp .F-ob A → CommRingHom A (Γ .F-ob X) fst (α ♯) a = α ●ᵛ yonedaᴾ 𝔸¹ A .inv a pres0 (snd (α ♯)) = makeNatTransPath (funExt₂ λ B x → α .N-ob B x .snd .pres0) @@ -187,53 +187,53 @@ -- the two maps are inverse to each other - ♭♯Id : ∀ (α : X ⇒ Sp .F-ob A) → ((α ♯) ♭) ≡ α + ♭♯Id : ∀ (α : X ⇒ Sp .F-ob A) → ((α ♯) ♭) ≡ α ♭♯Id _ = makeNatTransPath (funExt₂ λ _ _ → RingHom≡ (funExt (λ _ → refl))) - ♯♭Id : ∀ (φ : CommRingHom A (Γ .F-ob X)) → ((φ ♭) ♯) ≡ φ + ♯♭Id : ∀ (φ : CommRingHom A (Γ .F-ob X)) → ((φ ♭) ♯) ≡ φ ♯♭Id _ = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) Γ⊣SpIso : {A : CommRing ℓ} {X : ℤFunctor {ℓ}} - → Iso (CommRingHom A (Γ .F-ob X)) (X ⇒ Sp .F-ob A) + → Iso (CommRingHom A (Γ .F-ob X)) (X ⇒ Sp .F-ob A) fun Γ⊣SpIso = _♭ inv Γ⊣SpIso = _♯ rightInv Γ⊣SpIso = ♭♯Id leftInv Γ⊣SpIso = ♯♭Id - Γ⊣SpNatℤFunctor : {A : CommRing ℓ} {X Y : ℤFunctor {ℓ}} (α : X ⇒ Sp .F-ob A) (β : Y ⇒ X) - → (β ●ᵛ α) ♯ ≡ (Γ .F-hom β) ∘cr (α ♯) + Γ⊣SpNatℤFunctor : {A : CommRing ℓ} {X Y : ℤFunctor {ℓ}} (α : X ⇒ Sp .F-ob A) (β : Y ⇒ X) + → (β ●ᵛ α) ♯ ≡ (Γ .F-hom β) ∘cr (α ♯) Γ⊣SpNatℤFunctor _ _ = RingHom≡ (funExt (λ _ → makeNatTransPath (funExt₂ (λ _ _ → refl)))) Γ⊣SpNatCommRing : {X : ℤFunctor {ℓ}} {A B : CommRing ℓ} - (φ : CommRingHom A (Γ .F-ob X)) (ψ : CommRingHom B A) - → (φ ∘cr ψ) ♭ ≡ (φ ♭) ●ᵛ Sp .F-hom ψ + (φ : CommRingHom A (Γ .F-ob X)) (ψ : CommRingHom B A) + → (φ ∘cr ψ) ♭ ≡ (φ ♭) ●ᵛ Sp .F-hom ψ Γ⊣SpNatCommRing _ _ = makeNatTransPath (funExt₂ λ _ _ → RingHom≡ (funExt (λ _ → refl))) -- the counit is an equivalence private - ε : (A : CommRing ℓ) → CommRingHom A ((Γ ∘F Sp) .F-ob A) - ε A = (idTrans (Sp .F-ob A)) ♯ + ε : (A : CommRing ℓ) → CommRingHom A ((Γ ∘F Sp) .F-ob A) + ε A = (idTrans (Sp .F-ob A)) ♯ - Γ⊣SpCounitEquiv : (A : CommRing ℓ) → CommRingEquiv A ((Γ ∘F Sp) .F-ob A) + Γ⊣SpCounitEquiv : (A : CommRing ℓ) → CommRingEquiv A ((Γ ∘F Sp) .F-ob A) fst (Γ⊣SpCounitEquiv A) = isoToEquiv theIso where - theIso : Iso (A .fst) ((Γ ∘F Sp) .F-ob A .fst) + theIso : Iso (A .fst) ((Γ ∘F Sp) .F-ob A .fst) fun theIso = ε A .fst inv theIso = yonedaᴾ 𝔸¹ A .fun rightInv theIso α = ℤFUNCTOR .⋆IdL _ ∙ yonedaᴾ 𝔸¹ A .leftInv α leftInv theIso a = path -- I get yellow otherwise where - path : yonedaᴾ 𝔸¹ A .fun ((idTrans (Sp .F-ob A)) ●ᵛ yonedaᴾ 𝔸¹ A .inv a) ≡ a + path : yonedaᴾ 𝔸¹ A .fun ((idTrans (Sp .F-ob A)) ●ᵛ yonedaᴾ 𝔸¹ A .inv a) ≡ a path = cong (yonedaᴾ 𝔸¹ A .fun) (ℤFUNCTOR .⋆IdL _) ∙ yonedaᴾ 𝔸¹ A .rightInv a snd (Γ⊣SpCounitEquiv A) = ε A .snd -- Affine schemes module _ {ℓ : Level} where - open Functor + open Functor isAffine : (X : ℤFunctor) → Type (ℓ-suc ℓ) - isAffine X = ∃[ A ∈ CommRing ℓ ] NatIso (Sp .F-ob A) X + isAffine X = ∃[ A ∈ CommRing ℓ ] NatIso (Sp .F-ob A) X -- TODO: 𝔸¹ ≅ Sp ℤ[x] and 𝔾ₘ ≅ Sp ℤ[x,x⁻¹] as first examples of affine schemes @@ -244,11 +244,11 @@ -- We can however give terms that look just like the unit, -- giving us an alternative def. of affine ℤ-functors private module AffineDefs {ℓ : Level} where - open Functor + open Functor open NatTrans open Iso open IsRingHom - η : (X : ℤFunctor) (A : CommRing ℓ) → X .F-ob A .fst → CommRingHom (Γ .F-ob X) A + η : (X : ℤFunctor) (A : CommRing ℓ) → X .F-ob A .fst → CommRingHom (Γ .F-ob X) A fst (η X A x) α = α .N-ob A x pres0 (snd (η X A x)) = refl pres1 (snd (η X A x)) = refl @@ -258,12 +258,12 @@ -- this is basically a natural transformation ηObHom : (X : ℤFunctor) {A B : CommRing ℓ} (φ : CommRingHom A B) - → η X B ∘ (X .F-hom φ) ≡ (φ ∘cr_) ∘ η X A + → η X B ∘ (X .F-hom φ) ≡ (φ ∘cr_) ∘ η X A ηObHom X φ = funExt (λ x → RingHom≡ (funExt λ α → funExt⁻ (α .N-hom φ) x)) -- can only state equality on object part, but that would be enough - ηHom : {X Y : ℤFunctor} (α : X ⇒ Y) (A : CommRing ℓ) (x : X .F-ob A .fst) - → η Y A (α .N-ob A x) ≡ η X A x ∘cr Γ .F-hom α + ηHom : {X Y : ℤFunctor} (α : X ⇒ Y) (A : CommRing ℓ) (x : X .F-ob A .fst) + → η Y A (α .N-ob A x) ≡ η X A x ∘cr Γ .F-hom α ηHom _ _ _ = RingHom≡ refl isAffine' : (X : ℤFunctor) → Type (ℓ-suc ℓ) @@ -275,7 +275,7 @@ module _ {ℓ : Level} where open Iso - open Functor + open Functor open NatTrans open DistLatticeStr ⦃...⦄ open CommRingStr ⦃...⦄ @@ -286,34 +286,34 @@ -- the Zariski lattice classifying compact open subobjects 𝓛 : ℤFunctor {ℓ = ℓ} - F-ob 𝓛 A = ZL A , SQ.squash/ - F-hom 𝓛 φ = inducedZarLatHom φ .fst - F-id 𝓛 {A} = cong fst (inducedZarLatHomId A) - F-seq 𝓛 φ ψ = cong fst (inducedZarLatHomSeq φ ψ) + F-ob 𝓛 A = ZL A , SQ.squash/ + F-hom 𝓛 φ = inducedZarLatHom φ .fst + F-id 𝓛 {A} = cong fst (inducedZarLatHomId A) + F-seq 𝓛 φ ψ = cong fst (inducedZarLatHomSeq φ ψ) CompactOpen : ℤFunctor → Type (ℓ-suc ℓ) CompactOpen X = X ⇒ 𝓛 -- the induced subfunctor ⟦_⟧ᶜᵒ : {X : ℤFunctor} (U : CompactOpen X) → ℤFunctor - F-ob (⟦_⟧ᶜᵒ {X = X} U) A = (Σ[ x ∈ X .F-ob A .fst ] U .N-ob A x ≡ D A 1r) - , isSetΣSndProp (X .F-ob A .snd) λ _ → squash/ _ _ + F-ob (⟦_⟧ᶜᵒ {X = X} U) A = (Σ[ x ∈ X .F-ob A .fst ] U .N-ob A x ≡ D A 1r) + , isSetΣSndProp (X .F-ob A .snd) λ _ → squash/ _ _ where instance _ = snd A - F-hom (⟦_⟧ᶜᵒ {X = X} U) {x = A} {y = B} φ (x , Ux≡D1) = (X .F-hom φ x) , path + F-hom (⟦_⟧ᶜᵒ {X = X} U) {x = A} {y = B} φ (x , Ux≡D1) = (X .F-hom φ x) , path where instance _ = A .snd _ = B .snd open IsLatticeHom - path : U .N-ob B (X .F-hom φ x) ≡ D B 1r - path = U .N-ob B (X .F-hom φ x) ≡⟨ funExt⁻ (U .N-hom φ) x ⟩ - 𝓛 .F-hom φ (U .N-ob A x) ≡⟨ cong (𝓛 .F-hom φ) Ux≡D1 ⟩ - 𝓛 .F-hom φ (D A 1r) ≡⟨ inducedZarLatHom φ .snd .pres1 ⟩ + path : U .N-ob B (X .F-hom φ x) ≡ D B 1r + path = U .N-ob B (X .F-hom φ x) ≡⟨ funExt⁻ (U .N-hom φ) x ⟩ + 𝓛 .F-hom φ (U .N-ob A x) ≡⟨ cong (𝓛 .F-hom φ) Ux≡D1 ⟩ + 𝓛 .F-hom φ (D A 1r) ≡⟨ inducedZarLatHom φ .snd .pres1 ⟩ D B 1r ∎ - F-id (⟦_⟧ᶜᵒ {X = X} U) = funExt (λ x → Σ≡Prop (λ _ → squash/ _ _) - (funExt⁻ (X .F-id) (x .fst))) - F-seq (⟦_⟧ᶜᵒ {X = X} U) φ ψ = funExt (λ x → Σ≡Prop (λ _ → squash/ _ _) - (funExt⁻ (X .F-seq φ ψ) (x .fst))) + F-id (⟦_⟧ᶜᵒ {X = X} U) = funExt (λ x → Σ≡Prop (λ _ → squash/ _ _) + (funExt⁻ (X .F-id) (x .fst))) + F-seq (⟦_⟧ᶜᵒ {X = X} U) φ ψ = funExt (λ x → Σ≡Prop (λ _ → squash/ _ _) + (funExt⁻ (X .F-seq φ ψ) (x .fst))) isAffineCompactOpen : {X : ℤFunctor} (U : CompactOpen X) → Type (ℓ-suc ℓ) @@ -322,17 +322,17 @@ -- TODO: define basic opens D(f) ↪ Sp A and prove ⟦ D(f) ⟧ᶜᵒ ≅ Sp A[1/f] -- the (big) dist. lattice of compact opens - CompOpenDistLattice : Functor ℤFUNCTOR (DistLatticesCategory {ℓ = ℓ-suc ℓ} ^op) - fst (F-ob CompOpenDistLattice X) = CompactOpen X + CompOpenDistLattice : Functor ℤFUNCTOR (DistLatticesCategory {ℓ = ℓ-suc ℓ} ^op) + fst (F-ob CompOpenDistLattice X) = CompactOpen X -- lattice structure induce by internal lattice object 𝓛 - N-ob (DistLatticeStr.0l (snd (F-ob CompOpenDistLattice X))) A _ = 0l + N-ob (DistLatticeStr.0l (snd (F-ob CompOpenDistLattice X))) A _ = 0l where instance _ = ZariskiLattice A .snd - N-hom (DistLatticeStr.0l (snd (F-ob CompOpenDistLattice X))) _ = funExt λ _ → refl + N-hom (DistLatticeStr.0l (snd (F-ob CompOpenDistLattice X))) _ = funExt λ _ → refl - N-ob (DistLatticeStr.1l (snd (F-ob CompOpenDistLattice X))) A _ = 1l + N-ob (DistLatticeStr.1l (snd (F-ob CompOpenDistLattice X))) A _ = 1l where instance _ = ZariskiLattice A .snd - N-hom (DistLatticeStr.1l (snd (F-ob CompOpenDistLattice X))) {x = A} {y = B} φ = funExt λ _ → path + N-hom (DistLatticeStr.1l (snd (F-ob CompOpenDistLattice X))) {x = A} {y = B} φ = funExt λ _ → path where instance _ = A .snd @@ -341,38 +341,38 @@ path : D B 1r ≡ D B (φ .fst 1r) ∨l 0l path = cong (D B) (sym (φ .snd .pres1)) ∙ sym (∨lRid _) - N-ob ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∨l U) V) A x = U .N-ob A x ∨l V .N-ob A x + N-ob ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∨l U) V) A x = U .N-ob A x ∨l V .N-ob A x where instance _ = ZariskiLattice A .snd - N-hom ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∨l U) V) {x = A} {y = B} φ = funExt path + N-hom ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∨l U) V) {x = A} {y = B} φ = funExt path where instance _ = ZariskiLattice A .snd _ = ZariskiLattice B .snd - path : ∀ x → U .N-ob B (X .F-hom φ x) ∨l V .N-ob B (X .F-hom φ x) - ≡ 𝓛 .F-hom φ (U .N-ob A x ∨l V .N-ob A x) - path x = U .N-ob B (X .F-hom φ x) ∨l V .N-ob B (X .F-hom φ x) + path : ∀ x → U .N-ob B (X .F-hom φ x) ∨l V .N-ob B (X .F-hom φ x) + ≡ 𝓛 .F-hom φ (U .N-ob A x ∨l V .N-ob A x) + path x = U .N-ob B (X .F-hom φ x) ∨l V .N-ob B (X .F-hom φ x) ≡⟨ cong₂ _∨l_ (funExt⁻ (U .N-hom φ) x) (funExt⁻ (V .N-hom φ) x) ⟩ - 𝓛 .F-hom φ (U .N-ob A x) ∨l 𝓛 .F-hom φ (V .N-ob A x) + 𝓛 .F-hom φ (U .N-ob A x) ∨l 𝓛 .F-hom φ (V .N-ob A x) ≡⟨ sym (inducedZarLatHom φ .snd .pres∨l _ _) ⟩ - 𝓛 .F-hom φ (U .N-ob A x ∨l V .N-ob A x) ∎ + 𝓛 .F-hom φ (U .N-ob A x ∨l V .N-ob A x) ∎ - N-ob ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∧l U) V) A x = U .N-ob A x ∧l V .N-ob A x + N-ob ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∧l U) V) A x = U .N-ob A x ∧l V .N-ob A x where instance _ = ZariskiLattice A .snd - N-hom ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∧l U) V) {x = A} {y = B} φ = funExt path + N-hom ((snd (F-ob CompOpenDistLattice X) DistLatticeStr.∧l U) V) {x = A} {y = B} φ = funExt path where instance _ = ZariskiLattice A .snd _ = ZariskiLattice B .snd - path : ∀ x → U .N-ob B (X .F-hom φ x) ∧l V .N-ob B (X .F-hom φ x) - ≡ 𝓛 .F-hom φ (U .N-ob A x ∧l V .N-ob A x) - path x = U .N-ob B (X .F-hom φ x) ∧l V .N-ob B (X .F-hom φ x) + path : ∀ x → U .N-ob B (X .F-hom φ x) ∧l V .N-ob B (X .F-hom φ x) + ≡ 𝓛 .F-hom φ (U .N-ob A x ∧l V .N-ob A x) + path x = U .N-ob B (X .F-hom φ x) ∧l V .N-ob B (X .F-hom φ x) ≡⟨ cong₂ _∧l_ (funExt⁻ (U .N-hom φ) x) (funExt⁻ (V .N-hom φ) x) ⟩ - 𝓛 .F-hom φ (U .N-ob A x) ∧l 𝓛 .F-hom φ (V .N-ob A x) + 𝓛 .F-hom φ (U .N-ob A x) ∧l 𝓛 .F-hom φ (V .N-ob A x) ≡⟨ sym (inducedZarLatHom φ .snd .pres∧l _ _) ⟩ - 𝓛 .F-hom φ (U .N-ob A x ∧l V .N-ob A x) ∎ + 𝓛 .F-hom φ (U .N-ob A x ∧l V .N-ob A x) ∎ - DistLatticeStr.isDistLattice (snd (F-ob CompOpenDistLattice X)) = makeIsDistLattice∧lOver∨l - isSetNatTrans + DistLatticeStr.isDistLattice (snd (F-ob CompOpenDistLattice X)) = makeIsDistLattice∧lOver∨l + isSetNatTrans (λ _ _ _ → makeNatTransPath (funExt₂ (λ A _ → ZariskiLattice A .snd .DistLatticeStr.∨lAssoc _ _ _))) (λ _ → makeNatTransPath (funExt₂ (λ A _ → ZariskiLattice A .snd .DistLatticeStr.∨lRid _))) @@ -387,25 +387,25 @@ (λ A _ → ZariskiLattice A .snd .DistLatticeStr.∧l-dist-∨l _ _ _ .fst))) -- (contravariant) action on morphisms - fst (F-hom CompOpenDistLattice α) = α ●ᵛ_ - pres0 (snd (F-hom CompOpenDistLattice α)) = makeNatTransPath (funExt₂ λ _ _ → refl) - pres1 (snd (F-hom CompOpenDistLattice α)) = makeNatTransPath (funExt₂ λ _ _ → refl) - pres∨l (snd (F-hom CompOpenDistLattice α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) - pres∧l (snd (F-hom CompOpenDistLattice α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) + fst (F-hom CompOpenDistLattice α) = α ●ᵛ_ + pres0 (snd (F-hom CompOpenDistLattice α)) = makeNatTransPath (funExt₂ λ _ _ → refl) + pres1 (snd (F-hom CompOpenDistLattice α)) = makeNatTransPath (funExt₂ λ _ _ → refl) + pres∨l (snd (F-hom CompOpenDistLattice α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) + pres∧l (snd (F-hom CompOpenDistLattice α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) -- functoriality - F-id CompOpenDistLattice = LatticeHom≡f _ _ + F-id CompOpenDistLattice = LatticeHom≡f _ _ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) - F-seq CompOpenDistLattice _ _ = LatticeHom≡f _ _ + F-seq CompOpenDistLattice _ _ = LatticeHom≡f _ _ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) module _ (X : ℤFunctor) where - open Join (CompOpenDistLattice .F-ob X) - open JoinSemilattice (Lattice→JoinSemilattice (DistLattice→Lattice (CompOpenDistLattice .F-ob X))) + open Join (CompOpenDistLattice .F-ob X) + open JoinSemilattice (Lattice→JoinSemilattice (DistLattice→Lattice (CompOpenDistLattice .F-ob X))) open PosetStr (IndPoset .snd) hiding (_≤_) open LatticeTheory ⦃...⦄ -- ((DistLattice→Lattice (CompOpenDistLattice .F-ob X))) - private instance _ = (CompOpenDistLattice .F-ob X) .snd + private instance _ = (CompOpenDistLattice .F-ob X) .snd record AffineCover : Type (ℓ-suc ℓ) where field @@ -419,7 +419,7 @@ -- TODO: A ℤ-functor is a qcqs-scheme if it is a Zariski sheaf and has an affine cover -- the structure sheaf - private COᵒᵖ = (DistLatticeCategory (CompOpenDistLattice .F-ob X)) ^op + private COᵒᵖ = (DistLatticeCategory (CompOpenDistLattice .F-ob X)) ^op compOpenIncl : {U V : CompactOpen X} → V ≤ U → ⟦ V ⟧ᶜᵒ ⇒ ⟦ U ⟧ᶜᵒ N-ob (compOpenIncl {U = U} {V = V} V≤U) A (x , Vx≡D1) = x , path @@ -433,20 +433,20 @@ V .N-ob A x ∨l U .N-ob A x ≡⟨ cong (_∨l U .N-ob A x) Vx≡D1 ⟩ D A 1r ∨l U .N-ob A x ≡⟨ 1lLeftAnnihilates∨l _ ⟩ D A 1r ∎ - N-hom (compOpenIncl V≤U) φ = funExt λ x → Σ≡Prop (λ _ → squash/ _ _) refl + N-hom (compOpenIncl V≤U) φ = funExt λ x → Σ≡Prop (λ _ → squash/ _ _) refl compOpenInclId : ∀ {U : CompactOpen X} → compOpenIncl (is-refl U) ≡ idTrans ⟦ U ⟧ᶜᵒ - compOpenInclId = makeNatTransPath (funExt₂ (λ _ _ → Σ≡Prop (λ _ → squash/ _ _) refl)) + compOpenInclId = makeNatTransPath (funExt₂ (λ _ _ → Σ≡Prop (λ _ → squash/ _ _) refl)) compOpenInclSeq : ∀ {U V W : CompactOpen X} (U≤V : U ≤ V) (V≤W : V ≤ W) → compOpenIncl (is-trans _ _ _ U≤V V≤W) ≡ compOpenIncl U≤V ●ᵛ compOpenIncl V≤W compOpenInclSeq _ _ = makeNatTransPath - (funExt₂ (λ _ _ → Σ≡Prop (λ _ → squash/ _ _) refl)) + (funExt₂ (λ _ _ → Σ≡Prop (λ _ → squash/ _ _) refl)) - 𝓞 : Functor COᵒᵖ (CommRingsCategory {ℓ = ℓ-suc ℓ}) - F-ob 𝓞 U = Γ .F-ob ⟦ U ⟧ᶜᵒ - F-hom 𝓞 U≥V = Γ .F-hom (compOpenIncl U≥V) - F-id 𝓞 = cong (Γ .F-hom) compOpenInclId ∙ Γ .F-id - F-seq 𝓞 _ _ = cong (Γ .F-hom) (compOpenInclSeq _ _) ∙ Γ .F-seq _ _ + 𝓞 : Functor COᵒᵖ (CommRingsCategory {ℓ = ℓ-suc ℓ}) + F-ob 𝓞 U = Γ .F-ob ⟦ U ⟧ᶜᵒ + F-hom 𝓞 U≥V = Γ .F-hom (compOpenIncl U≥V) + F-id 𝓞 = cong (Γ .F-hom) compOpenInclId ∙ Γ .F-id + F-seq 𝓞 _ _ = cong (Γ .F-hom) (compOpenInclSeq _ _) ∙ Γ .F-seq _ _ \ No newline at end of file diff --git a/Cubical.Categories.Isomorphism.html b/Cubical.Categories.Isomorphism.html index 7081585fdf..03fb5f2cd4 100644 --- a/Cubical.Categories.Isomorphism.html +++ b/Cubical.Categories.Isomorphism.html @@ -207,38 +207,38 @@ op-Iso f .snd .ret = f .snd .ret -module _ {C : Category ℓC ℓC'}{D : Category ℓD ℓD'}{F : Functor C D} where +module _ {C : Category ℓC ℓC'}{D : Category ℓD ℓD'}{F : Functor C D} where open Category hiding (_∘_) open isIso - open Functor + open Functor - F-PresIsIso : {x y : C .ob}{f : C [ x , y ]} → isIso C f → isIso D (F .F-hom f) - F-PresIsIso f .inv = F . F-hom (f .inv) - F-PresIsIso f .sec = sym (F .F-seq (f .inv) _) ∙ cong (F .F-hom) (f .sec) ∙ F .F-id - F-PresIsIso f .ret = sym (F .F-seq _ (f .inv)) ∙ cong (F .F-hom) (f .ret) ∙ F .F-id + F-PresIsIso : {x y : C .ob}{f : C [ x , y ]} → isIso C f → isIso D (F .F-hom f) + F-PresIsIso f .inv = F . F-hom (f .inv) + F-PresIsIso f .sec = sym (F .F-seq (f .inv) _) ∙ cong (F .F-hom) (f .sec) ∙ F .F-id + F-PresIsIso f .ret = sym (F .F-seq _ (f .inv)) ∙ cong (F .F-hom) (f .ret) ∙ F .F-id - F-Iso : {x y : C .ob} → CatIso C x y → CatIso D (F .F-ob x) (F .F-ob y) - F-Iso f .fst = F . F-hom (f .fst) + F-Iso : {x y : C .ob} → CatIso C x y → CatIso D (F .F-ob x) (F .F-ob y) + F-Iso f .fst = F . F-hom (f .fst) F-Iso f .snd = F-PresIsIso (f .snd) F-Iso-PresId : {x : C .ob} → F-Iso (idCatIso {x = x}) ≡ idCatIso - F-Iso-PresId = CatIso≡ _ _ (F .F-id) + F-Iso-PresId = CatIso≡ _ _ (F .F-id) F-Iso-Pres⋆ : {x y z : C .ob} → (f : CatIso C x y)(g : CatIso C y z) → F-Iso (⋆Iso f g) ≡ ⋆Iso (F-Iso f) (F-Iso g) - F-Iso-Pres⋆ _ _ = CatIso≡ _ _ (F .F-seq _ _) + F-Iso-Pres⋆ _ _ = CatIso≡ _ _ (F .F-seq _ _) - F-pathToIso : {x y : C .ob} → (p : x ≡ y) → F-Iso (pathToIso p) ≡ pathToIso (cong (F .F-ob) p) - F-pathToIso p = J (λ y p → F-Iso (pathToIso p) ≡ pathToIso (cong (F .F-ob) p)) F-pathToIso-refl p + F-pathToIso : {x y : C .ob} → (p : x ≡ y) → F-Iso (pathToIso p) ≡ pathToIso (cong (F .F-ob) p) + F-pathToIso p = J (λ y p → F-Iso (pathToIso p) ≡ pathToIso (cong (F .F-ob) p)) F-pathToIso-refl p where - F-pathToIso-refl : {x : C .ob} → F-Iso (pathToIso {x = x} refl) ≡ pathToIso (cong (F .F-ob) refl) + F-pathToIso-refl : {x : C .ob} → F-Iso (pathToIso {x = x} refl) ≡ pathToIso (cong (F .F-ob) refl) F-pathToIso-refl = cong F-Iso pathToIso-refl ∙ F-Iso-PresId ∙ sym pathToIso-refl - F-pathToIso-∘ : {x y : C .ob} → F-Iso ∘ pathToIso {x = x} {y = y} ≡ pathToIso ∘ cong (F .F-ob) + F-pathToIso-∘ : {x y : C .ob} → F-Iso ∘ pathToIso {x = x} {y = y} ≡ pathToIso ∘ cong (F .F-ob) F-pathToIso-∘ i p = F-pathToIso p i \ No newline at end of file diff --git a/Cubical.Categories.Limits.Initial.html b/Cubical.Categories.Limits.Initial.html index 347c5bd42c..5ecf7b8287 100644 --- a/Cubical.Categories.Limits.Initial.html +++ b/Cubical.Categories.Limits.Initial.html @@ -63,24 +63,24 @@ -- i.e. all initial objects are equal. isPropInitial : (hC : isUnivalent C) → isProp Initial isPropInitial hC x y = - Σ≡Prop isPropIsInitial (CatIsoToPath hC (initialToIso x y)) + Σ≡Prop isPropIsInitial (CatIsoToPath hC (initialToIso x y)) -module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) where +module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} (F : Functor C D) where open Category - open Functor - open NaturalBijection - open _⊣_ + open Functor + open NaturalBijection + open _⊣_ open _≅_ preservesInitial : Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) - preservesInitial = ∀ (x : ob C) → isInitial C x → isInitial D (F-ob F x) + preservesInitial = ∀ (x : ob C) → isInitial C x → isInitial D (F-ob F x) - isLeftAdjoint→preservesInitial : isLeftAdjoint F → preservesInitial - fst (isLeftAdjoint→preservesInitial (G , F⊣G) x initX y) = _♯ F⊣G (fst (initX (F-ob G y))) + isLeftAdjoint→preservesInitial : isLeftAdjoint F → preservesInitial + fst (isLeftAdjoint→preservesInitial (G , F⊣G) x initX y) = _♯ F⊣G (fst (initX (F-ob G y))) snd (isLeftAdjoint→preservesInitial (G , F⊣G) x initX y) ψ = - _♯ F⊣G (fst (initX (F-ob G y))) - ≡⟨ cong (F⊣G ♯) (snd (initX (F-ob G y)) (_♭ F⊣G ψ)) ⟩ - _♯ F⊣G (_♭ F⊣G ψ) - ≡⟨ leftInv (adjIso F⊣G) ψ ⟩ + _♯ F⊣G (fst (initX (F-ob G y))) + ≡⟨ cong (F⊣G ♯) (snd (initX (F-ob G y)) (_♭ F⊣G ψ)) ⟩ + _♯ F⊣G (_♭ F⊣G ψ) + ≡⟨ leftInv (adjIso F⊣G) ψ ⟩ ψ ∎ \ No newline at end of file diff --git a/Cubical.Categories.Limits.Limits.html b/Cubical.Categories.Limits.Limits.html index 2f989620a0..c71b778285 100644 --- a/Cubical.Categories.Limits.Limits.html +++ b/Cubical.Categories.Limits.Limits.html @@ -23,59 +23,59 @@ module _ {ℓJ ℓJ' ℓC ℓC' : Level} {J : Category ℓJ ℓJ'} {C : Category ℓC ℓC'} where open Category - open Functor + open Functor open NatTrans private ℓ = ℓ-max (ℓ-max (ℓ-max ℓJ ℓJ') ℓC) ℓC' - record Cone (D : Functor J C) (c : ob C) : Type (ℓ-max (ℓ-max ℓJ ℓJ') ℓC') where + record Cone (D : Functor J C) (c : ob C) : Type (ℓ-max (ℓ-max ℓJ ℓJ') ℓC') where constructor cone field - coneOut : (v : ob J) → C [ c , F-ob D v ] + coneOut : (v : ob J) → C [ c , F-ob D v ] coneOutCommutes : {u v : ob J} (e : J [ u , v ]) → - coneOut u ⋆⟨ C ⟩ D .F-hom e ≡ coneOut v + coneOut u ⋆⟨ C ⟩ D .F-hom e ≡ coneOut v open Cone - cone≡ : {D : Functor J C} {c : ob C} → {c1 c2 : Cone D c} + cone≡ : {D : Functor J C} {c : ob C} → {c1 c2 : Cone D c} → ((v : ob J) → coneOut c1 v ≡ coneOut c2 v) → c1 ≡ c2 coneOut (cone≡ h i) v = h v i coneOutCommutes (cone≡ {D} {_} {c1} {c2} h i) {u} {v} e = - isProp→PathP (λ j → isSetHom C (h u j ⋆⟨ C ⟩ D .F-hom e) (h v j)) + isProp→PathP (λ j → isSetHom C (h u j ⋆⟨ C ⟩ D .F-hom e) (h v j)) (coneOutCommutes c1 e) (coneOutCommutes c2 e) i -- dependent versions - conePathP : {D₁ D₂ : Functor J C} {c₁ c₂ : ob C} {cc₁ : Cone D₁ c₁} {cc₂ : Cone D₂ c₂} + conePathP : {D₁ D₂ : Functor J C} {c₁ c₂ : ob C} {cc₁ : Cone D₁ c₁} {cc₂ : Cone D₂ c₂} {p : D₁ ≡ D₂} {q : c₁ ≡ c₂} - → (∀ v → PathP (λ i → C [ q i , p i .F-ob v ]) (cc₁ .coneOut v) (cc₂ .coneOut v)) + → (∀ v → PathP (λ i → C [ q i , p i .F-ob v ]) (cc₁ .coneOut v) (cc₂ .coneOut v)) → PathP (λ i → Cone (p i) (q i)) cc₁ cc₂ coneOut (conePathP coneOutPathP i) v = coneOutPathP v i coneOutCommutes (conePathP {cc₁ = cc₁} {cc₂ = cc₂} {p = p} coneOutPathP i) {u} {v} e = - isProp→PathP (λ j → isSetHom C (coneOutPathP u j ⋆⟨ C ⟩ p j .F-hom e) + isProp→PathP (λ j → isSetHom C (coneOutPathP u j ⋆⟨ C ⟩ p j .F-hom e) (coneOutPathP v j)) (coneOutCommutes cc₁ e) (coneOutCommutes cc₂ e) i - conePathPOb : {D : Functor J C} {c c' : ob C} {c1 : Cone D c} {c2 : Cone D c'} {p : c ≡ c'} - → (∀ (v : ob J) → PathP (λ i → C [ p i , F-ob D v ]) (coneOut c1 v) (coneOut c2 v)) + conePathPOb : {D : Functor J C} {c c' : ob C} {c1 : Cone D c} {c2 : Cone D c'} {p : c ≡ c'} + → (∀ (v : ob J) → PathP (λ i → C [ p i , F-ob D v ]) (coneOut c1 v) (coneOut c2 v)) → PathP (λ i → Cone D (p i)) c1 c2 conePathPOb coneOutPathP = conePathP {p = refl} coneOutPathP - conePathPDiag : {D₁ D₂ : Functor J C} {c : ob C} {cc₁ : Cone D₁ c} {cc₂ : Cone D₂ c} {p : D₁ ≡ D₂} - → (∀ v → PathP (λ i → C [ c , p i .F-ob v ]) (cc₁ .coneOut v) (cc₂ .coneOut v)) + conePathPDiag : {D₁ D₂ : Functor J C} {c : ob C} {cc₁ : Cone D₁ c} {cc₂ : Cone D₂ c} {p : D₁ ≡ D₂} + → (∀ v → PathP (λ i → C [ c , p i .F-ob v ]) (cc₁ .coneOut v) (cc₂ .coneOut v)) → PathP (λ i → Cone (p i) c) cc₁ cc₂ conePathPDiag coneOutPathP = conePathP {q = refl} coneOutPathP -- TODO: can we automate this a bit? - isSetCone : (D : Functor J C) (c : ob C) → isSet (Cone D c) + isSetCone : (D : Functor J C) (c : ob C) → isSet (Cone D c) isSetCone D c = isSetRetract toConeΣ fromConeΣ (λ _ → refl) (isSetΣSndProp (isSetΠ (λ _ → isSetHom C)) (λ _ → isPropImplicitΠ2 (λ _ _ → isPropΠ (λ _ → isSetHom C _ _)))) where - ConeΣ = Σ[ f ∈ ((v : ob J) → C [ c , F-ob D v ]) ] - ({u v : ob J} (e : J [ u , v ]) → f u ⋆⟨ C ⟩ D .F-hom e ≡ f v) + ConeΣ = Σ[ f ∈ ((v : ob J) → C [ c , F-ob D v ]) ] + ({u v : ob J} (e : J [ u , v ]) → f u ⋆⟨ C ⟩ D .F-hom e ≡ f v) toConeΣ : Cone D c → ConeΣ fst (toConeΣ x) = coneOut x @@ -85,72 +85,72 @@ coneOut (fromConeΣ x) = fst x coneOutCommutes (fromConeΣ x) = snd x - preCompCone : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} + preCompCone : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} → Cone D c2 → Cone D c1 coneOut (preCompCone f cc) v = f ⋆⟨ C ⟩ coneOut cc v coneOutCommutes (preCompCone f cc) e = ⋆Assoc C _ _ _ ∙ cong (λ x → f ⋆⟨ C ⟩ x) (coneOutCommutes cc e) - _★_ : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} → Cone D c2 → Cone D c1 + _★_ : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} → Cone D c2 → Cone D c1 _★_ = preCompCone - natTransPostCompCone : {c : ob C} {D₁ D₂ : Functor J C} (α : NatTrans D₁ D₂) + natTransPostCompCone : {c : ob C} {D₁ D₂ : Functor J C} (α : NatTrans D₁ D₂) → Cone D₁ c → Cone D₂ c coneOut (natTransPostCompCone α cc) u = cc .coneOut u ⋆⟨ C ⟩ α .N-ob u coneOutCommutes (natTransPostCompCone {D₁ = D₁} {D₂ = D₂} α cc) {u = u} {v = v} e = - cc .coneOut u ⋆⟨ C ⟩ α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e + cc .coneOut u ⋆⟨ C ⟩ α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e ≡⟨ ⋆Assoc C _ _ _ ⟩ - cc .coneOut u ⋆⟨ C ⟩ (α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e) + cc .coneOut u ⋆⟨ C ⟩ (α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e) ≡⟨ cong (λ x → cc .coneOut u ⋆⟨ C ⟩ x) (sym (α .N-hom e)) ⟩ - cc .coneOut u ⋆⟨ C ⟩ (D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v) + cc .coneOut u ⋆⟨ C ⟩ (D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - cc .coneOut u ⋆⟨ C ⟩ D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v + cc .coneOut u ⋆⟨ C ⟩ D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v ≡⟨ cong (λ x → x ⋆⟨ C ⟩ α .N-ob v) (cc .coneOutCommutes e) ⟩ cc .coneOut v ⋆⟨ C ⟩ α .N-ob v ∎ - _★ₙₜ_ : {c : ob C} {D₁ D₂ : Functor J C} → Cone D₁ c → NatTrans D₁ D₂ → Cone D₂ c + _★ₙₜ_ : {c : ob C} {D₁ D₂ : Functor J C} → Cone D₁ c → NatTrans D₁ D₂ → Cone D₂ c _★ₙₜ_ = flip natTransPostCompCone - isConeMor : {c1 c2 : ob C} {D : Functor J C} + isConeMor : {c1 c2 : ob C} {D : Functor J C} (cc1 : Cone D c1) (cc2 : Cone D c2) → C [ c1 , c2 ] → Type (ℓ-max ℓJ ℓC') isConeMor cc1 cc2 f = (v : ob J) → f ⋆⟨ C ⟩ coneOut cc2 v ≡ coneOut cc1 v - isPropIsConeMor : {c1 c2 : ob C} {D : Functor J C} + isPropIsConeMor : {c1 c2 : ob C} {D : Functor J C} (cc1 : Cone D c1) (cc2 : Cone D c2) (f : C [ c1 , c2 ]) → isProp (isConeMor cc1 cc2 f) isPropIsConeMor cc1 cc2 f = isPropΠ (λ _ → isSetHom C _ _) - isConeMorId : {c : ob C} {D : Functor J C} (cc : Cone D c) + isConeMorId : {c : ob C} {D : Functor J C} (cc : Cone D c) → isConeMor cc cc (id C) isConeMorId _ _ = ⋆IdL C _ - isConeMorSeq : {c1 c2 c3 : ob C} {D : Functor J C} + isConeMorSeq : {c1 c2 c3 : ob C} {D : Functor J C} (cc1 : Cone D c1) (cc2 : Cone D c2) (cc3 : Cone D c3) {f : C [ c1 , c2 ]} {g : C [ c2 , c3 ]} → isConeMor cc1 cc2 f → isConeMor cc2 cc3 g → isConeMor cc1 cc3 (f ⋆⟨ C ⟩ g) isConeMorSeq cc1 cc2 cc3 {f} {g} isConeMorF isConeMorG v = ⋆Assoc C _ _ _ ∙∙ cong (λ x → f ⋆⟨ C ⟩ x) (isConeMorG v) ∙∙ isConeMorF v - preCompConeMor : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} (cc : Cone D c2) + preCompConeMor : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} (cc : Cone D c2) → isConeMor (f ★ cc) cc f preCompConeMor f cc v = refl - isLimCone : (D : Functor J C) (c0 : ob C) → Cone D c0 → Type ℓ + isLimCone : (D : Functor J C) (c0 : ob C) → Cone D c0 → Type ℓ isLimCone D c0 cc0 = (c : ob C) → (cc : Cone D c) → ∃![ f ∈ C [ c , c0 ] ] isConeMor cc cc0 f - isPropIsLimCone : (D : Functor J C) (c0 : ob C) (cc0 : Cone D c0) + isPropIsLimCone : (D : Functor J C) (c0 : ob C) (cc0 : Cone D c0) → isProp (isLimCone D c0 cc0) isPropIsLimCone D c0 cc0 = isPropΠ2 λ _ _ → isProp∃! - preCompUnique : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} (cc : Cone D c2) + preCompUnique : {c1 c2 : ob C} (f : C [ c1 , c2 ]) {D : Functor J C} (cc : Cone D c2) → isLimCone D c2 cc → ∀ (g : C [ c1 , c2 ]) → isConeMor (f ★ cc) cc g → g ≡ f preCompUnique f cc ccIsLimCone g gIsConeMor = cong fst (isContr→isProp (ccIsLimCone _ (f ★ cc)) (g , gIsConeMor) (f , preCompConeMor f cc)) - record LimCone (D : Functor J C) : Type ℓ where + record LimCone (D : Functor J C) : Type ℓ where constructor climCone field @@ -158,11 +158,11 @@ limCone : Cone D lim univProp : isLimCone D lim limCone - limOut : (v : ob J) → C [ lim , D .F-ob v ] + limOut : (v : ob J) → C [ lim , D .F-ob v ] limOut = coneOut limCone limOutCommutes : {u v : ob J} (e : J [ u , v ]) - → limOut u ⋆⟨ C ⟩ D .F-hom e ≡ limOut v + → limOut u ⋆⟨ C ⟩ D .F-hom e ≡ limOut v limOutCommutes = coneOutCommutes limCone limArrow : (c : ob C) → (cc : Cone D c) → C [ c , lim ] @@ -177,35 +177,35 @@ limArrowUnique c cc k hk = cong fst (univProp c cc .snd (k , hk)) open LimCone - limOfArrowsCone : {D₁ D₂ : Functor J C} + limOfArrowsCone : {D₁ D₂ : Functor J C} (CC₁ : LimCone D₁) → NatTrans D₁ D₂ → Cone D₂ (CC₁ .lim) coneOut (limOfArrowsCone {D₁} {D₂} CC₁ α) v = limOut CC₁ v ⋆⟨ C ⟩ α .N-ob v coneOutCommutes (limOfArrowsCone {D₁} {D₂} CC₁ α) {u = u} {v = v} e = - limOut CC₁ u ⋆⟨ C ⟩ α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e ≡⟨ ⋆Assoc C _ _ _ ⟩ - limOut CC₁ u ⋆⟨ C ⟩ (α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e) ≡⟨ cong (λ x → seq' C (limOut CC₁ u) x) (sym (α .N-hom e)) ⟩ - limOut CC₁ u ⋆⟨ C ⟩ (D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ - limOut CC₁ u ⋆⟨ C ⟩ D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v ≡⟨ cong (λ x → x ⋆⟨ C ⟩ α .N-ob v) (limOutCommutes CC₁ e) ⟩ + limOut CC₁ u ⋆⟨ C ⟩ α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e ≡⟨ ⋆Assoc C _ _ _ ⟩ + limOut CC₁ u ⋆⟨ C ⟩ (α .N-ob u ⋆⟨ C ⟩ D₂ .F-hom e) ≡⟨ cong (λ x → seq' C (limOut CC₁ u) x) (sym (α .N-hom e)) ⟩ + limOut CC₁ u ⋆⟨ C ⟩ (D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v) ≡⟨ sym (⋆Assoc C _ _ _) ⟩ + limOut CC₁ u ⋆⟨ C ⟩ D₁ .F-hom e ⋆⟨ C ⟩ α .N-ob v ≡⟨ cong (λ x → x ⋆⟨ C ⟩ α .N-ob v) (limOutCommutes CC₁ e) ⟩ limOut CC₁ v ⋆⟨ C ⟩ α .N-ob v ∎ - limOfArrows : {D₁ D₂ : Functor J C} + limOfArrows : {D₁ D₂ : Functor J C} (CC₁ : LimCone D₁) (CC₂ : LimCone D₂) → NatTrans D₁ D₂ → C [ CC₁ .lim , CC₂ .lim ] limOfArrows {D₁} {D₂} CC₁ CC₂ α = limArrow CC₂ (CC₁ .lim) (limOfArrowsCone CC₁ α) - limOfArrowsOut : {D₁ D₂ : Functor J C} + limOfArrowsOut : {D₁ D₂ : Functor J C} (CC₁ : LimCone D₁) (CC₂ : LimCone D₂) (α : NatTrans D₁ D₂) (u : ob J) → limOfArrows CC₁ CC₂ α ⋆⟨ C ⟩ limOut CC₂ u ≡ limOut CC₁ u ⋆⟨ C ⟩ α .N-ob u limOfArrowsOut _ CC₂ _ _ = limArrowCommutes CC₂ _ _ _ - limOfArrowsId : {D : Functor J C} (CC : LimCone D) + limOfArrowsId : {D : Functor J C} (CC : LimCone D) → limOfArrows CC CC (idTrans D) ≡ id C limOfArrowsId CC = limArrowUnique CC _ _ _ λ v → ⋆IdL C _ ∙ sym (⋆IdR C _) - limOfArrowsSeq : {D₁ D₂ D₃ : Functor J C} + limOfArrowsSeq : {D₁ D₂ D₃ : Functor J C} (CC₁ : LimCone D₁) (CC₂ : LimCone D₂) (CC₃ : LimCone D₃) (α : NatTrans D₁ D₂) (β : NatTrans D₂ D₃) → limOfArrows CC₁ CC₃ (α ●ᵛ β) @@ -227,7 +227,7 @@ ≡⟨ ⋆Assoc C _ _ _ ⟩ limOut CC₁ u ⋆⟨ C ⟩ (α .N-ob u ⋆⟨ C ⟩ β .N-ob u) ∎ - limArrowCompLimOfArrows : {D₁ D₂ : Functor J C} + limArrowCompLimOfArrows : {D₁ D₂ : Functor J C} (CC₁ : LimCone D₁) (CC₂ : LimCone D₂) (α : NatTrans D₁ D₂) (c : ob C) (cc : Cone D₁ c) @@ -250,7 +250,7 @@ -- any two limits are isomorphic up to a unique cone isomorphism open isIso - LimIso : (D : Functor J C) (L₁ L₂ : LimCone D) + LimIso : (D : Functor J C) (L₁ L₂ : LimCone D) → ∃![ e ∈ CatIso C (lim L₁) (lim L₂) ] isConeMor (limCone L₁) (limCone L₂) (fst e) fst (fst (fst (LimIso D L₁ L₂))) = limArrow L₂ _ (limCone L₁) inv (snd (fst (fst (LimIso D L₁ L₂)))) = limArrow L₁ _ (limCone L₂) @@ -290,31 +290,31 @@ coneOut (limCone L₁) v ∎ snd (fst (LimIso D L₁ L₂)) = limArrowCommutes L₂ _ _ - snd (LimIso D L₁ L₂) (e , isConeMorE) = Σ≡Prop + snd (LimIso D L₁ L₂) (e , isConeMorE) = Σ≡Prop (λ _ → isPropIsConeMor (limCone L₁) (limCone L₂) _) (CatIso≡ _ _ (limArrowUnique L₂ _ _ (fst e) isConeMorE)) -- if the index category of the diagram has an initial object, -- we get a canonical limiting cone - Initial→LimCone : (D : Functor J C) → Initial J → LimCone D - lim (Initial→LimCone D (j , isInitJ)) = D .F-ob j - coneOut (limCone (Initial→LimCone D (j , isInitJ))) k = D .F-hom (isInitJ k .fst) + Initial→LimCone : (D : Functor J C) → Initial J → LimCone D + lim (Initial→LimCone D (j , isInitJ)) = D .F-ob j + coneOut (limCone (Initial→LimCone D (j , isInitJ))) k = D .F-hom (isInitJ k .fst) coneOutCommutes (limCone (Initial→LimCone D (j , isInitJ))) f = - sym (D .F-seq _ _) ∙ cong (D .F-hom) (sym (isInitJ _ .snd _)) + sym (D .F-seq _ _) ∙ cong (D .F-hom) (sym (isInitJ _ .snd _)) fst (fst (univProp (Initial→LimCone D (j , isInitJ)) c cc)) = cc .coneOut j -- canonical arrow c → D(j) snd (fst (univProp (Initial→LimCone D (j , isInitJ)) c cc)) k = cc .coneOutCommutes (isInitJ k .fst) -- is a cone morphism snd (univProp (Initial→LimCone D (j , isInitJ)) c cc) (f , isConeMorF) = -- and indeed unique - Σ≡Prop + Σ≡Prop (λ _ → isPropIsConeMor cc (limCone (Initial→LimCone D (j , isInitJ))) _) - (sym (isConeMorF j) ∙∙ cong (λ x → f ⋆⟨ C ⟩ x) (subst (λ x → D .F-hom x ≡ id C) - (sym (isInitJ j .snd _)) (D .F-id)) + (sym (isConeMorF j) ∙∙ cong (λ x → f ⋆⟨ C ⟩ x) (subst (λ x → D .F-hom x ≡ id C) + (sym (isInitJ j .snd _)) (D .F-id)) ∙∙ ⋆IdR C f) -- cones that respect isomorphisms are limiting cones - Iso→LimCone : {D : Functor J C} {c₁ c₂ : ob C} (cc₁ : Cone D c₁) (e : CatIso C c₁ c₂) + Iso→LimCone : {D : Functor J C} {c₁ c₂ : ob C} (cc₁ : Cone D c₁) (e : CatIso C c₁ c₂) → isLimCone _ _ cc₁ → (cc₂ : Cone D c₂) → isConeMor cc₁ cc₂ (e .fst) @@ -324,7 +324,7 @@ snd (fst (Iso→LimCone cc₁ e isLimConeCC₁ cc₂ isConeMorE d cd)) = isConeMorSeq cd cc₁ cc₂ (isLimConeCC₁ d cd .fst .snd) isConeMorE snd (Iso→LimCone cc₁ e isLimConeCC₁ cc₂ isConeMorE d cd) (f , isConeMorF) = - Σ≡Prop (isPropIsConeMor cd cc₂) path + Σ≡Prop (isPropIsConeMor cd cc₂) path where isConeMorE⁻¹ : isConeMor cc₂ cc₁ (e .snd .inv) isConeMorE⁻¹ v = @@ -348,60 +348,60 @@ -- A category is complete if it has all limits Limits : {ℓJ ℓJ' ℓC ℓC' : Level} → Category ℓC ℓC' → Type _ -Limits {ℓJ} {ℓJ'} C = (J : Category ℓJ ℓJ') → (D : Functor J C) → LimCone D +Limits {ℓJ} {ℓJ'} C = (J : Category ℓJ ℓJ') → (D : Functor J C) → LimCone D hasLimits : {ℓJ ℓJ' ℓC ℓC' : Level} → Category ℓC ℓC' → Type _ -hasLimits {ℓJ} {ℓJ'} C = (J : Category ℓJ ℓJ') → (D : Functor J C) → ∥ LimCone D ∥₁ +hasLimits {ℓJ} {ℓJ'} C = (J : Category ℓJ ℓJ') → (D : Functor J C) → ∥ LimCone D ∥₁ -- Limits of a specific shape J in a category C LimitsOfShape : {ℓJ ℓJ' ℓC ℓC' : Level} → Category ℓJ ℓJ' → Category ℓC ℓC' → Type _ -LimitsOfShape J C = (D : Functor J C) → LimCone D +LimitsOfShape J C = (D : Functor J C) → LimCone D -- precomposition with a functor and preservation of limits module _ {ℓJ ℓJ' ℓC1 ℓC1' ℓC2 ℓC2' : Level} {C1 : Category ℓC1 ℓC1'} {C2 : Category ℓC2 ℓC2'} - (F : Functor C1 C2) where + (F : Functor C1 C2) where open Category - open Functor + open Functor open Cone -- same as F-cone!!! - F-cone : {J : Category ℓJ ℓJ'} {D : Functor J C1} {x : ob C1} - → Cone D x → Cone (funcComp F D) (F .F-ob x) - coneOut (F-cone ccx) v = F .F-hom (coneOut ccx v) + F-cone : {J : Category ℓJ ℓJ'} {D : Functor J C1} {x : ob C1} + → Cone D x → Cone (funcComp F D) (F .F-ob x) + coneOut (F-cone ccx) v = F .F-hom (coneOut ccx v) coneOutCommutes (F-cone ccx) e = - sym (F-seq F (coneOut ccx _) _) ∙ cong (F .F-hom) (coneOutCommutes ccx e) + sym (F-seq F (coneOut ccx _) _) ∙ cong (F .F-hom) (coneOutCommutes ccx e) preservesLimits : Type _ - preservesLimits = ∀ {J : Category ℓJ ℓJ'} {D : Functor J C1} {L : ob C1} + preservesLimits = ∀ {J : Category ℓJ ℓJ'} {D : Functor J C1} {L : ob C1} (ccL : Cone D L) - → isLimCone _ _ ccL → isLimCone (funcComp F D) (F .F-ob L) (F-cone ccL) + → isLimCone _ _ ccL → isLimCone (funcComp F D) (F .F-ob L) (F-cone ccL) -- characterizing limit preserving functors open LimCone preservesLimitsChar : (L₁ : Limits {ℓJ} {ℓJ'} C1) (L₂ : Limits {ℓJ} {ℓJ'} C2) - → (e : ∀ J D → CatIso C2 (L₂ J (F ∘F D) .lim) (F .F-ob (L₁ J D .lim))) - → (∀ J D → isConeMor (L₂ J (F ∘F D) .limCone) (F-cone (L₁ J D .limCone)) (e J D .fst)) + → (e : ∀ J D → CatIso C2 (L₂ J (F ∘F D) .lim) (F .F-ob (L₁ J D .lim))) + → (∀ J D → isConeMor (L₂ J (F ∘F D) .limCone) (F-cone (L₁ J D .limCone)) (e J D .fst)) → preservesLimits preservesLimitsChar L₁ L₂ e isConeMorE {J = J} {D = D} {L = c} cc isLimConeCC = - Iso→LimCone (L₂ J (F ∘F D) .limCone) f (L₂ J (F ∘F D) .univProp) (F-cone cc) isConeMorF + Iso→LimCone (L₂ J (F ∘F D) .limCone) f (L₂ J (F ∘F D) .univProp) (F-cone cc) isConeMorF where theCLimCone : LimCone D lim theCLimCone = c limCone theCLimCone = cc univProp theCLimCone = isLimConeCC - f : CatIso C2 (lim (L₂ J (funcComp F D))) (F .F-ob c) + f : CatIso C2 (lim (L₂ J (funcComp F D))) (F .F-ob c) f = ⋆Iso (e J D) (F-Iso {F = F} (LimIso D (L₁ J D) theCLimCone .fst .fst)) - isConeMorF : isConeMor (L₂ J (funcComp F D) .limCone) (F-cone cc) (f .fst) - isConeMorF = isConeMorSeq (L₂ J (funcComp F D) .limCone) + isConeMorF : isConeMor (L₂ J (funcComp F D) .limCone) (F-cone cc) (f .fst) + isConeMorF = isConeMorSeq (L₂ J (funcComp F D) .limCone) (F-cone (L₁ J D .limCone)) (F-cone cc) (isConeMorE J D) - (λ v → F-triangle F (limArrowCommutes theCLimCone _ _ _)) + (λ v → F-triangle F (limArrowCommutes theCLimCone _ _ _)) -- TODO: prove that right adjoints preserve limits \ No newline at end of file diff --git a/Cubical.Categories.Limits.Pullback.html b/Cubical.Categories.Limits.Pullback.html index 8574e97288..aedb902b35 100644 --- a/Cubical.Categories.Limits.Pullback.html +++ b/Cubical.Categories.Limits.Pullback.html @@ -21,7 +21,7 @@ module _ (C : Category ℓ ℓ') where open Category C - open Functor + open Functor record Cospan : Type (ℓ-max ℓ ℓ') where constructor cospan @@ -96,31 +96,31 @@ -- Pullbacks from limits module _ {C : Category ℓ ℓ'} where open Category C - open Functor + open Functor open Pullback open LimCone open Cone open Cospan - Cospan→Func : Cospan C → Functor CospanCat C - Cospan→Func (cospan l m r f g) .F-ob ⓪ = l - Cospan→Func (cospan l m r f g) .F-ob ① = m - Cospan→Func (cospan l m r f g) .F-ob ② = r - Cospan→Func (cospan l m r f g) .F-hom {⓪} {①} k = f - Cospan→Func (cospan l m r f g) .F-hom {②} {①} k = g - Cospan→Func (cospan l m r f g) .F-hom {⓪} {⓪} k = id - Cospan→Func (cospan l m r f g) .F-hom {①} {①} k = id - Cospan→Func (cospan l m r f g) .F-hom {②} {②} k = id - Cospan→Func (cospan l m r f g) .F-id {⓪} = refl - Cospan→Func (cospan l m r f g) .F-id {①} = refl - Cospan→Func (cospan l m r f g) .F-id {②} = refl - Cospan→Func (cospan l m r f g) .F-seq {⓪} {⓪} {⓪} φ ψ = sym (⋆IdL _) - Cospan→Func (cospan l m r f g) .F-seq {⓪} {⓪} {①} φ ψ = sym (⋆IdL _) - Cospan→Func (cospan l m r f g) .F-seq {⓪} {①} {①} φ ψ = sym (⋆IdR _) - Cospan→Func (cospan l m r f g) .F-seq {①} {①} {①} φ ψ = sym (⋆IdL _) - Cospan→Func (cospan l m r f g) .F-seq {②} {②} {②} φ ψ = sym (⋆IdL _) - Cospan→Func (cospan l m r f g) .F-seq {②} {②} {①} φ ψ = sym (⋆IdL _) - Cospan→Func (cospan l m r f g) .F-seq {②} {①} {①} φ ψ = sym (⋆IdR _) + Cospan→Func : Cospan C → Functor CospanCat C + Cospan→Func (cospan l m r f g) .F-ob ⓪ = l + Cospan→Func (cospan l m r f g) .F-ob ① = m + Cospan→Func (cospan l m r f g) .F-ob ② = r + Cospan→Func (cospan l m r f g) .F-hom {⓪} {①} k = f + Cospan→Func (cospan l m r f g) .F-hom {②} {①} k = g + Cospan→Func (cospan l m r f g) .F-hom {⓪} {⓪} k = id + Cospan→Func (cospan l m r f g) .F-hom {①} {①} k = id + Cospan→Func (cospan l m r f g) .F-hom {②} {②} k = id + Cospan→Func (cospan l m r f g) .F-id {⓪} = refl + Cospan→Func (cospan l m r f g) .F-id {①} = refl + Cospan→Func (cospan l m r f g) .F-id {②} = refl + Cospan→Func (cospan l m r f g) .F-seq {⓪} {⓪} {⓪} φ ψ = sym (⋆IdL _) + Cospan→Func (cospan l m r f g) .F-seq {⓪} {⓪} {①} φ ψ = sym (⋆IdL _) + Cospan→Func (cospan l m r f g) .F-seq {⓪} {①} {①} φ ψ = sym (⋆IdR _) + Cospan→Func (cospan l m r f g) .F-seq {①} {①} {①} φ ψ = sym (⋆IdL _) + Cospan→Func (cospan l m r f g) .F-seq {②} {②} {②} φ ψ = sym (⋆IdL _) + Cospan→Func (cospan l m r f g) .F-seq {②} {②} {①} φ ψ = sym (⋆IdL _) + Cospan→Func (cospan l m r f g) .F-seq {②} {①} {①} φ ψ = sym (⋆IdR _) LimitsOfShapeCospanCat→Pullbacks : LimitsOfShape CospanCat C → Pullbacks C pbOb (LimitsOfShapeCospanCat→Pullbacks H cspn) = lim (H (Cospan→Func cspn)) diff --git a/Cubical.Categories.Limits.RightKan.html b/Cubical.Categories.Limits.RightKan.html index cbcba3dcae..2c5b98825e 100644 --- a/Cubical.Categories.Limits.RightKan.html +++ b/Cubical.Categories.Limits.RightKan.html @@ -28,160 +28,160 @@ {M : Category ℓM ℓM'} {A : Category ℓA ℓA'} (limitA : Limits {ℓ-max ℓC' ℓM} {ℓ-max ℓC' ℓM'} A) - (K : Functor M C) - (T : Functor M A) + (K : Functor M C) + (T : Functor M A) where open Category - open Functor + open Functor open Cone open LimCone _↓Diag : ob C → Category (ℓ-max ℓC' ℓM) (ℓ-max ℓC' ℓM') - ob (x ↓Diag) = Σ[ u ∈ ob M ] C [ x , K .F-ob u ] - Hom[_,_] (x ↓Diag) (u , f) (v , g) = Σ[ h ∈ M [ u , v ] ] f ⋆⟨ C ⟩ K .F-hom h ≡ g - id (x ↓Diag) {x = (u , f)} = id M , cong (seq' C f) (F-id K) ∙ ⋆IdR C f + ob (x ↓Diag) = Σ[ u ∈ ob M ] C [ x , K .F-ob u ] + Hom[_,_] (x ↓Diag) (u , f) (v , g) = Σ[ h ∈ M [ u , v ] ] f ⋆⟨ C ⟩ K .F-hom h ≡ g + id (x ↓Diag) {x = (u , f)} = id M , cong (seq' C f) (F-id K) ∙ ⋆IdR C f _⋆_ (x ↓Diag) {x = (u , f)} (h , hComm) (k , kComm) = (h ⋆⟨ M ⟩ k) - , cong (seq' C f) (F-seq K h k) + , cong (seq' C f) (F-seq K h k) ∙ sym (⋆Assoc C _ _ _) - ∙ cong (λ l → seq' C l (F-hom K k)) hComm + ∙ cong (λ l → seq' C l (F-hom K k)) hComm ∙ kComm - ⋆IdL (x ↓Diag) _ = Σ≡Prop (λ _ → isSetHom C _ _) (⋆IdL M _) - ⋆IdR (x ↓Diag) _ = Σ≡Prop (λ _ → isSetHom C _ _) (⋆IdR M _) - ⋆Assoc (x ↓Diag) _ _ _ = Σ≡Prop (λ _ → isSetHom C _ _) (⋆Assoc M _ _ _) + ⋆IdL (x ↓Diag) _ = Σ≡Prop (λ _ → isSetHom C _ _) (⋆IdL M _) + ⋆IdR (x ↓Diag) _ = Σ≡Prop (λ _ → isSetHom C _ _) (⋆IdR M _) + ⋆Assoc (x ↓Diag) _ _ _ = Σ≡Prop (λ _ → isSetHom C _ _) (⋆Assoc M _ _ _) isSetHom (x ↓Diag) = isSetΣSndProp (isSetHom M) λ _ → isSetHom C _ _ private - i : (x : ob C) → Functor (x ↓Diag) M - F-ob (i x) = fst - F-hom (i x) = fst - F-id (i x) = refl - F-seq (i x) _ _ = refl + i : (x : ob C) → Functor (x ↓Diag) M + F-ob (i x) = fst + F-hom (i x) = fst + F-id (i x) = refl + F-seq (i x) _ _ = refl - j : {x y : ob C} (f : C [ x , y ]) → Functor (y ↓Diag) (x ↓Diag) - F-ob (j f) (u , g) = u , f ⋆⟨ C ⟩ g - F-hom (j f) (h , hComm) = h , ⋆Assoc C _ _ _ ∙ cong (seq' C f) hComm - F-id (j f) = Σ≡Prop (λ _ → isSetHom C _ _) refl - F-seq (j f) _ _ = Σ≡Prop (λ _ → isSetHom C _ _) refl + j : {x y : ob C} (f : C [ x , y ]) → Functor (y ↓Diag) (x ↓Diag) + F-ob (j f) (u , g) = u , f ⋆⟨ C ⟩ g + F-hom (j f) (h , hComm) = h , ⋆Assoc C _ _ _ ∙ cong (seq' C f) hComm + F-id (j f) = Σ≡Prop (λ _ → isSetHom C _ _) refl + F-seq (j f) _ _ = Σ≡Prop (λ _ → isSetHom C _ _) refl - T* : (x : ob C) → Functor (x ↓Diag) A - T* x = funcComp T (i x) + T* : (x : ob C) → Functor (x ↓Diag) A + T* x = funcComp T (i x) RanOb : ob C → ob A RanOb x = limitA (x ↓Diag) (T* x) .lim RanCone : {x y : ob C} → C [ x , y ] → Cone (T* y) (RanOb x) - coneOut (RanCone {x = x} f) v = limOut (limitA (x ↓Diag) (T* x)) (j f .F-ob v) - coneOutCommutes (RanCone {x = x} f) h = limOutCommutes (limitA (x ↓Diag) (T* x)) (j f .F-hom h) + coneOut (RanCone {x = x} f) v = limOut (limitA (x ↓Diag) (T* x)) (j f .F-ob v) + coneOutCommutes (RanCone {x = x} f) h = limOutCommutes (limitA (x ↓Diag) (T* x)) (j f .F-hom h) private -- technical lemmas for proving functoriality RanConeRefl : ∀ {x} v → limOut (limitA (x ↓Diag) (T* x)) v - ≡ limOut (limitA (x ↓Diag) (T* x)) (j (id C) .F-ob v) + ≡ limOut (limitA (x ↓Diag) (T* x)) (j (id C) .F-ob v) RanConeRefl {x = x} (v , f) = cong (λ p → limOut (limitA (x ↓Diag) (T* x)) (v , p)) (sym (⋆IdL C f)) RanConeTrans : ∀ {x y z} (f : C [ x , y ]) (g : C [ y , z ]) v → - limOut (limitA (x ↓Diag) (T* x)) (j f .F-ob (j g .F-ob v)) - ≡ limOut (limitA (x ↓Diag) (T* x)) (j (f ⋆⟨ C ⟩ g) .F-ob v) + limOut (limitA (x ↓Diag) (T* x)) (j f .F-ob (j g .F-ob v)) + ≡ limOut (limitA (x ↓Diag) (T* x)) (j (f ⋆⟨ C ⟩ g) .F-ob v) RanConeTrans {x = x} {y = y} {z = z} f g (v , h) = cong (λ p → limOut (limitA (x ↓Diag) (T* x)) (v , p)) (sym (⋆Assoc C f g h)) -- the right Kan-extension - Ran : Functor C A - F-ob Ran = RanOb - F-hom Ran {y = y} f = limArrow (limitA (y ↓Diag) (T* y)) _ (RanCone f) - F-id Ran {x = x} = + Ran : Functor C A + F-ob Ran = RanOb + F-hom Ran {y = y} f = limArrow (limitA (y ↓Diag) (T* y)) _ (RanCone f) + F-id Ran {x = x} = limArrowUnique (limitA (x ↓Diag) (T* x)) _ _ _ (λ v → (⋆IdL A _) ∙ RanConeRefl v) - F-seq Ran {x = x} {y = y} {z = z} f g = + F-seq Ran {x = x} {y = y} {z = z} f g = limArrowUnique (limitA (z ↓Diag) (T* z)) _ _ _ path where path : ∀ v → - (F-hom Ran f) ⋆⟨ A ⟩ (F-hom Ran g) ⋆⟨ A ⟩ (limOut (limitA (z ↓Diag) (T* z)) v) + (F-hom Ran f) ⋆⟨ A ⟩ (F-hom Ran g) ⋆⟨ A ⟩ (limOut (limitA (z ↓Diag) (T* z)) v) ≡ coneOut (RanCone (f ⋆⟨ C ⟩ g)) v - path v = (F-hom Ran f) ⋆⟨ A ⟩ (F-hom Ran g) ⋆⟨ A ⟩ (limOut (limitA (z ↓Diag) (T* z)) v) + path v = (F-hom Ran f) ⋆⟨ A ⟩ (F-hom Ran g) ⋆⟨ A ⟩ (limOut (limitA (z ↓Diag) (T* z)) v) ≡⟨ ⋆Assoc A _ _ _ ⟩ - (F-hom Ran f) ⋆⟨ A ⟩ ((F-hom Ran g) ⋆⟨ A ⟩ (limOut (limitA (z ↓Diag) (T* z)) v)) - ≡⟨ cong (seq' A (F-hom Ran f)) (limArrowCommutes _ _ _ _) ⟩ - (F-hom Ran f) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (j g .F-ob v) + (F-hom Ran f) ⋆⟨ A ⟩ ((F-hom Ran g) ⋆⟨ A ⟩ (limOut (limitA (z ↓Diag) (T* z)) v)) + ≡⟨ cong (seq' A (F-hom Ran f)) (limArrowCommutes _ _ _ _) ⟩ + (F-hom Ran f) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (j g .F-ob v) ≡⟨ limArrowCommutes _ _ _ _ ⟩ - limOut (limitA (x ↓Diag) (T* x)) (j f .F-ob (j g .F-ob v)) + limOut (limitA (x ↓Diag) (T* x)) (j f .F-ob (j g .F-ob v)) ≡⟨ RanConeTrans f g v ⟩ coneOut (RanCone (f ⋆⟨ C ⟩ g)) v ∎ open NatTrans - RanNatTrans : NatTrans (funcComp Ran K) T + RanNatTrans : NatTrans (funcComp Ran K) T N-ob RanNatTrans u = coneOut (RanCone (id C)) (u , id C) N-hom RanNatTrans {x = u} {y = v} f = - Ran .F-hom (K .F-hom f) ⋆⟨ A ⟩ coneOut (RanCone (id C)) (v , id C) - ≡⟨ cong (λ g → Ran .F-hom (K .F-hom f) ⋆⟨ A ⟩ g) (sym (RanConeRefl (v , id C))) ⟩ - Ran .F-hom (K .F-hom f) ⋆⟨ A ⟩ limOut (limitA ((K .F-ob v) ↓Diag) (T* (K .F-ob v))) (v , id C) + Ran .F-hom (K .F-hom f) ⋆⟨ A ⟩ coneOut (RanCone (id C)) (v , id C) + ≡⟨ cong (λ g → Ran .F-hom (K .F-hom f) ⋆⟨ A ⟩ g) (sym (RanConeRefl (v , id C))) ⟩ + Ran .F-hom (K .F-hom f) ⋆⟨ A ⟩ limOut (limitA ((K .F-ob v) ↓Diag) (T* (K .F-ob v))) (v , id C) ≡⟨ limArrowCommutes _ _ _ _ ⟩ - coneOut (RanCone (K .F-hom f)) (v , id C) - ≡⟨ cong (λ g → limOut (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) (v , g)) - (⋆IdR C (K .F-hom f) ∙ sym (⋆IdL C (K .F-hom f))) ⟩ - coneOut (RanCone (id C)) (v , K .F-hom f) + coneOut (RanCone (K .F-hom f)) (v , id C) + ≡⟨ cong (λ g → limOut (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) (v , g)) + (⋆IdR C (K .F-hom f) ∙ sym (⋆IdL C (K .F-hom f))) ⟩ + coneOut (RanCone (id C)) (v , K .F-hom f) ≡⟨ sym (coneOutCommutes (RanCone (id C)) (f , ⋆IdL C _)) ⟩ - coneOut (RanCone (id C)) (u , id C) ⋆⟨ A ⟩ T .F-hom f ∎ + coneOut (RanCone (id C)) (u , id C) ⋆⟨ A ⟩ T .F-hom f ∎ -- the universal property private ε = RanNatTrans - NatTransCone : (S : Functor C A) (α : NatTrans (S ∘F K) T) (x : ob C) - → Cone (T* x) (S .F-ob x) - coneOut (NatTransCone S α x) (v , f) = S .F-hom f ⋆⟨ A ⟩ α .N-ob v + NatTransCone : (S : Functor C A) (α : NatTrans (S ∘F K) T) (x : ob C) + → Cone (T* x) (S .F-ob x) + coneOut (NatTransCone S α x) (v , f) = S .F-hom f ⋆⟨ A ⟩ α .N-ob v coneOutCommutes (NatTransCone S α x) {u = (u , f)} {v = (v , g)} (h , f⋆Kh≡g) = - (S .F-hom f ⋆⟨ A ⟩ α .N-ob u) ⋆⟨ A ⟩ T .F-hom h + (S .F-hom f ⋆⟨ A ⟩ α .N-ob u) ⋆⟨ A ⟩ T .F-hom h ≡⟨ ⋆Assoc A _ _ _ ⟩ - S .F-hom f ⋆⟨ A ⟩ (α .N-ob u ⋆⟨ A ⟩ T .F-hom h) - ≡⟨ cong (seq' A (S .F-hom f)) (sym (α .N-hom h)) ⟩ - S .F-hom f ⋆⟨ A ⟩ ((S ∘F K) .F-hom h ⋆⟨ A ⟩ α .N-ob v) + S .F-hom f ⋆⟨ A ⟩ (α .N-ob u ⋆⟨ A ⟩ T .F-hom h) + ≡⟨ cong (seq' A (S .F-hom f)) (sym (α .N-hom h)) ⟩ + S .F-hom f ⋆⟨ A ⟩ ((S ∘F K) .F-hom h ⋆⟨ A ⟩ α .N-ob v) ≡⟨ sym (⋆Assoc A _ _ _) ⟩ - (S .F-hom f ⋆⟨ A ⟩ (S ∘F K) .F-hom h) ⋆⟨ A ⟩ α .N-ob v - ≡⟨ cong (λ x → x ⋆⟨ A ⟩ (α .N-ob v)) (sym (S .F-seq _ _)) ⟩ - S .F-hom (f ⋆⟨ C ⟩ K .F-hom h) ⋆⟨ A ⟩ α .N-ob v - ≡⟨ cong (λ x → S .F-hom x ⋆⟨ A ⟩ (α .N-ob v)) f⋆Kh≡g ⟩ - S .F-hom g ⋆⟨ A ⟩ α .N-ob v ∎ + (S .F-hom f ⋆⟨ A ⟩ (S ∘F K) .F-hom h) ⋆⟨ A ⟩ α .N-ob v + ≡⟨ cong (λ x → x ⋆⟨ A ⟩ (α .N-ob v)) (sym (S .F-seq _ _)) ⟩ + S .F-hom (f ⋆⟨ C ⟩ K .F-hom h) ⋆⟨ A ⟩ α .N-ob v + ≡⟨ cong (λ x → S .F-hom x ⋆⟨ A ⟩ (α .N-ob v)) f⋆Kh≡g ⟩ + S .F-hom g ⋆⟨ A ⟩ α .N-ob v ∎ - RanUnivProp : ∀ (S : Functor C A) (α : NatTrans (S ∘F K) T) + RanUnivProp : ∀ (S : Functor C A) (α : NatTrans (S ∘F K) T) → ∃![ σ ∈ NatTrans S Ran ] α ≡ (σ ∘ˡ K) ●ᵛ ε - RanUnivProp S α = uniqueExists σ pathNatTrans (λ _ → isSetNatTrans _ _) pathUnique + RanUnivProp S α = uniqueExists σ pathNatTrans (λ _ → isSetNatTrans _ _) pathUnique where σ : NatTrans S Ran N-ob σ x = limArrow (limitA (x ↓Diag) (T* x)) _ (NatTransCone S α x) - N-hom σ {x = x} {y = y} f = preCompUnique (N-ob σ x ⋆⟨ A ⟩ Ran .F-hom f) + N-hom σ {x = x} {y = y} f = preCompUnique (N-ob σ x ⋆⟨ A ⟩ Ran .F-hom f) (limitA (y ↓Diag) (T* y) .limCone) (limitA (y ↓Diag) (T* y) .univProp) - (S .F-hom f ⋆⟨ A ⟩ N-ob σ y) + (S .F-hom f ⋆⟨ A ⟩ N-ob σ y) equalUpToConeOut where equalUpToConeOut : ∀ (u : ob (y ↓Diag)) - → (S .F-hom f ⋆⟨ A ⟩ N-ob σ y) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) u - ≡ (N-ob σ x ⋆⟨ A ⟩ Ran .F-hom f) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) u + → (S .F-hom f ⋆⟨ A ⟩ N-ob σ y) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) u + ≡ (N-ob σ x ⋆⟨ A ⟩ Ran .F-hom f) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) u equalUpToConeOut (u , g) = - (S .F-hom f ⋆⟨ A ⟩ N-ob σ y) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g) + (S .F-hom f ⋆⟨ A ⟩ N-ob σ y) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g) ≡⟨ ⋆Assoc A _ _ _ ⟩ - S .F-hom f ⋆⟨ A ⟩ (N-ob σ y ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g)) - ≡⟨ cong (seq' A (S .F-hom f)) (limArrowCommutes _ _ _ _) ⟩ - S .F-hom f ⋆⟨ A ⟩ (S .F-hom g ⋆⟨ A ⟩ α .N-ob u) + S .F-hom f ⋆⟨ A ⟩ (N-ob σ y ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g)) + ≡⟨ cong (seq' A (S .F-hom f)) (limArrowCommutes _ _ _ _) ⟩ + S .F-hom f ⋆⟨ A ⟩ (S .F-hom g ⋆⟨ A ⟩ α .N-ob u) ≡⟨ sym (⋆Assoc A _ _ _) ⟩ - (S .F-hom f ⋆⟨ A ⟩ S .F-hom g) ⋆⟨ A ⟩ α .N-ob u - ≡⟨ cong (λ h → h ⋆⟨ A ⟩ α .N-ob u) (sym (S .F-seq _ _)) ⟩ - S .F-hom (f ⋆⟨ C ⟩ g) ⋆⟨ A ⟩ α .N-ob u + (S .F-hom f ⋆⟨ A ⟩ S .F-hom g) ⋆⟨ A ⟩ α .N-ob u + ≡⟨ cong (λ h → h ⋆⟨ A ⟩ α .N-ob u) (sym (S .F-seq _ _)) ⟩ + S .F-hom (f ⋆⟨ C ⟩ g) ⋆⟨ A ⟩ α .N-ob u ≡⟨ sym (limArrowCommutes _ _ _ _) ⟩ N-ob σ x ⋆⟨ A ⟩ limOut (limitA (x ↓Diag) (T* x)) (u , f ⋆⟨ C ⟩ g) ≡⟨ cong (seq' A (N-ob σ x)) (sym (limArrowCommutes _ _ _ _)) ⟩ - N-ob σ x ⋆⟨ A ⟩ (Ran .F-hom f ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g)) + N-ob σ x ⋆⟨ A ⟩ (Ran .F-hom f ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g)) ≡⟨ sym (⋆Assoc A _ _ _) ⟩ - (N-ob σ x ⋆⟨ A ⟩ Ran .F-hom f) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g) ∎ + (N-ob σ x ⋆⟨ A ⟩ Ran .F-hom f) ⋆⟨ A ⟩ limOut (limitA (y ↓Diag) (T* y)) (u , g) ∎ pathNatTrans : α ≡ (σ ∘ˡ K) ●ᵛ ε @@ -189,21 +189,21 @@ where path : ∀ (u : ob M) → α .N-ob u - ≡ limArrow (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) _ (NatTransCone S α (F-ob K u)) - ⋆⟨ A ⟩ limOut (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) (u , id C ⋆⟨ C ⟩ id C) + ≡ limArrow (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) _ (NatTransCone S α (F-ob K u)) + ⋆⟨ A ⟩ limOut (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) (u , id C ⋆⟨ C ⟩ id C) path u = α .N-ob u ≡⟨ sym (⋆IdL A _) ⟩ id A ⋆⟨ A ⟩ α .N-ob u - ≡⟨ cong (λ x → x ⋆⟨ A ⟩ α .N-ob u) (sym (S .F-id)) ⟩ - S .F-hom (id C) ⋆⟨ A ⟩ α .N-ob u - ≡⟨ cong (λ x → S .F-hom x ⋆⟨ A ⟩ α .N-ob u) (sym (⋆IdL C _)) ⟩ - S .F-hom (id C ⋆⟨ C ⟩ id C) ⋆⟨ A ⟩ α .N-ob u + ≡⟨ cong (λ x → x ⋆⟨ A ⟩ α .N-ob u) (sym (S .F-id)) ⟩ + S .F-hom (id C) ⋆⟨ A ⟩ α .N-ob u + ≡⟨ cong (λ x → S .F-hom x ⋆⟨ A ⟩ α .N-ob u) (sym (⋆IdL C _)) ⟩ + S .F-hom (id C ⋆⟨ C ⟩ id C) ⋆⟨ A ⟩ α .N-ob u ≡⟨ refl ⟩ - NatTransCone S α (F-ob K u) .coneOut (u , id C ⋆⟨ C ⟩ id C) + NatTransCone S α (F-ob K u) .coneOut (u , id C ⋆⟨ C ⟩ id C) ≡⟨ sym (limArrowCommutes _ _ _ _) ⟩ - limArrow (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) _ (NatTransCone S α (F-ob K u)) - ⋆⟨ A ⟩ limOut (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) (u , id C ⋆⟨ C ⟩ id C) ∎ + limArrow (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) _ (NatTransCone S α (F-ob K u)) + ⋆⟨ A ⟩ limOut (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) (u , id C ⋆⟨ C ⟩ id C) ∎ pathUnique : ∀ (τ : NatTrans S Ran) → α ≡ (τ ∘ˡ K) ●ᵛ ε → σ ≡ τ pathUnique τ α≡ετK = makeNatTransPath @@ -219,17 +219,17 @@ ≡⟨ (λ i → τ .N-ob x ⋆⟨ A ⟩ limOut (limitA (x ↓Diag) (T* x)) (u , p i)) ⟩ τ .N-ob x ⋆⟨ A ⟩ limOut (limitA (x ↓Diag) (T* x)) (u , f ⋆⟨ C ⟩ (id C ⋆⟨ C ⟩ id C)) ≡⟨ cong (seq' A (τ .N-ob x)) (sym (limArrowCommutes - (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) + (limitA ((K .F-ob u) ↓Diag) (T* (K .F-ob u))) _ _ _)) ⟩ - τ .N-ob x ⋆⟨ A ⟩ (Ran .F-hom f ⋆⟨ A ⟩ ε .N-ob u) + τ .N-ob x ⋆⟨ A ⟩ (Ran .F-hom f ⋆⟨ A ⟩ ε .N-ob u) ≡⟨ sym (⋆Assoc A _ _ _) ⟩ - (τ .N-ob x ⋆⟨ A ⟩ Ran .F-hom f) ⋆⟨ A ⟩ ε .N-ob u + (τ .N-ob x ⋆⟨ A ⟩ Ran .F-hom f) ⋆⟨ A ⟩ ε .N-ob u ≡⟨ cong (λ g → g ⋆⟨ A ⟩ ε .N-ob u) (sym (τ .N-hom f)) ⟩ - (S .F-hom f ⋆⟨ A ⟩ τ .N-ob (K .F-ob u)) ⋆⟨ A ⟩ ε .N-ob u + (S .F-hom f ⋆⟨ A ⟩ τ .N-ob (K .F-ob u)) ⋆⟨ A ⟩ ε .N-ob u ≡⟨ ⋆Assoc A _ _ _ ⟩ - S .F-hom f ⋆⟨ A ⟩ (τ .N-ob (K .F-ob u) ⋆⟨ A ⟩ ε .N-ob u) - ≡⟨ cong (seq' A (S .F-hom f)) (cong (λ γ → γ .N-ob u) (sym α≡ετK)) ⟩ - S .F-hom f ⋆⟨ A ⟩ α .N-ob u ∎ + S .F-hom f ⋆⟨ A ⟩ (τ .N-ob (K .F-ob u) ⋆⟨ A ⟩ ε .N-ob u) + ≡⟨ cong (seq' A (S .F-hom f)) (cong (λ γ → γ .N-ob u) (sym α≡ετK)) ⟩ + S .F-hom f ⋆⟨ A ⟩ α .N-ob u ∎ @@ -237,25 +237,25 @@ open isIso open NatIso - RanNatIso : isFullyFaithful K → NatIso (funcComp Ran K) T + RanNatIso : isFullyFaithful K → NatIso (funcComp Ran K) T trans (RanNatIso isFFK) = RanNatTrans - nIso (RanNatIso isFFK) u = LimIso _ (limitA (K .F-ob u ↓Diag) (T* (K .F-ob u))) + nIso (RanNatIso isFFK) u = LimIso _ (limitA (K .F-ob u ↓Diag) (T* (K .F-ob u))) (Initial→LimCone _ uInitial) .fst .fst .snd where - invKHom : {u v : ob M} → C [ K .F-ob u , K .F-ob v ] → M [ u , v ] - invKHom f = invEq (K .F-hom , (isFFK _ _)) f + invKHom : {u v : ob M} → C [ K .F-ob u , K .F-ob v ] → M [ u , v ] + invKHom f = invEq (K .F-hom , (isFFK _ _)) f - secKHom : ∀ {u v : ob M} (f : C [ K .F-ob u , K .F-ob v ]) - → K .F-hom (invKHom f) ≡ f - secKHom f = secEq (K .F-hom , (isFFK _ _)) f + secKHom : ∀ {u v : ob M} (f : C [ K .F-ob u , K .F-ob v ]) + → K .F-hom (invKHom f) ≡ f + secKHom f = secEq (K .F-hom , (isFFK _ _)) f - uInitial : Initial (K .F-ob u ↓Diag) + uInitial : Initial (K .F-ob u ↓Diag) fst uInitial = u , id C ⋆⟨ C ⟩ id C fst (snd uInitial (v , f)) = invKHom f -- the unique arrow u→v in Ku↓ , cong₂ (seq' C) (⋆IdL C _) (secKHom f) ∙ ⋆IdL C f - snd (snd uInitial (v , f)) (g , tr) = Σ≡Prop (λ _ → isSetHom C _ _) path -- is indeed unique + snd (snd uInitial (v , f)) (g , tr) = Σ≡Prop (λ _ → isSetHom C _ _) path -- is indeed unique where path : invKHom f ≡ g path = isFullyFaithful→Faithful {F = K} isFFK _ _ _ _ - (secKHom f ∙∙ sym tr ∙∙ cong (λ x → x ⋆⟨ C ⟩ K .F-hom g) (⋆IdL C _) ∙ ⋆IdL C _) + (secKHom f ∙∙ sym tr ∙∙ cong (λ x → x ⋆⟨ C ⟩ K .F-hom g) (⋆IdL C _) ∙ ⋆IdL C _) \ No newline at end of file diff --git a/Cubical.Categories.Limits.Terminal.html b/Cubical.Categories.Limits.Terminal.html index 1ff6756231..e93375e5c7 100644 --- a/Cubical.Categories.Limits.Terminal.html +++ b/Cubical.Categories.Limits.Terminal.html @@ -67,19 +67,19 @@ -- i.e. all terminal objects are equal. isPropTerminal : (hC : isUnivalent C) → isProp Terminal isPropTerminal hC x y = - Σ≡Prop isPropIsTerminal (CatIsoToPath hC (terminalToIso x y)) + Σ≡Prop isPropIsTerminal (CatIsoToPath hC (terminalToIso x y)) preservesTerminals : ∀ (C : Category ℓc ℓc')(D : Category ℓd ℓd') - → Functor C D + → Functor C D → Type (ℓ-max (ℓ-max (ℓ-max ℓc ℓc') ℓd) ℓd') -preservesTerminals C D F = ∀ (term : Terminal C) → isTerminal D (F ⟅ term .fst ⟆) +preservesTerminals C D F = ∀ (term : Terminal C) → isTerminal D (F ⟅ term .fst ⟆) preserveAnyTerminal→PreservesTerminals : ∀ (C : Category ℓc ℓc')(D : Category ℓd ℓd') - → (F : Functor C D) - → (term : Terminal C) → isTerminal D (F ⟅ term .fst ⟆) + → (F : Functor C D) + → (term : Terminal C) → isTerminal D (F ⟅ term .fst ⟆) → preservesTerminals C D F preserveAnyTerminal→PreservesTerminals C D F term D-preserves-term term' = isoToTerminal D - ((F ⟅ term .fst ⟆) , D-preserves-term) (F ⟅ term' .fst ⟆) + ((F ⟅ term .fst ⟆) , D-preserves-term) (F ⟅ term' .fst ⟆) (F-Iso {F = F} (terminalToIso C term term')) \ No newline at end of file diff --git a/Cubical.Categories.Monad.Base.html b/Cubical.Categories.Monad.Base.html index 3b633ac1de..4606e7fc4f 100644 --- a/Cubical.Categories.Monad.Base.html +++ b/Cubical.Categories.Monad.Base.html @@ -4,7 +4,7 @@ open import Cubical.Foundations.Prelude open import Cubical.Categories.Category -open import Cubical.Categories.Functor renaming (𝟙⟨_⟩ to funcId) +open import Cubical.Categories.Functor renaming (𝟙⟨_⟩ to funcId) open import Cubical.Categories.NaturalTransformation.Base open import Cubical.Categories.NaturalTransformation.Properties open import Cubical.Categories.Functors.HomFunctor @@ -14,9 +14,9 @@ variable ℓ ℓ' : Level -module _ {C : Category ℓ ℓ'} (M : Functor C C) where +module _ {C : Category ℓ ℓ'} (M : Functor C C) where open Category C - open Functor + open Functor open NatTrans IsPointed : Type (ℓ-max ℓ ℓ') @@ -25,7 +25,7 @@ record IsMonad : Type (ℓ-max ℓ ℓ') where field η : IsPointed - μ : NatTrans (funcComp M M) M + μ : NatTrans (funcComp M M) M idl-μ : PathP (λ i → NatTrans (F-rUnit {F = M} i) M) (compTrans μ (η ∘ˡ M)) (idTrans M) idr-μ : PathP (λ i → NatTrans (F-lUnit {F = M} i) M) (compTrans μ (M ∘ʳ η)) (idTrans M) assoc-μ : PathP (λ i → NatTrans (F-assoc {F = M} {G = M} {H = M} i) M) @@ -33,33 +33,33 @@ (compTrans μ (μ ∘ˡ M)) -- bind : Hom[-,M-] -> Hom[M-,M-] - bind : NatTrans (funcComp (HomFunctor C) ((funcId C ^opF) ×F M)) (funcComp (HomFunctor C) ((M ^opF) ×F M)) - N-ob bind (x , y) f = N-ob μ y ∘ F-hom M f + bind : NatTrans (funcComp (HomFunctor C) ((funcId C ^opF) ×F M)) (funcComp (HomFunctor C) ((M ^opF) ×F M)) + N-ob bind (x , y) f = N-ob μ y ∘ F-hom M f N-hom bind {x , y} {x' , y'} (f , h) = funExt λ g → - (F-hom M ((f ⋆ g) ⋆ F-hom M h) ⋆ N-ob μ y') - ≡⟨ cong (_⋆ N-ob μ y') (F-seq M (f ⋆ g) (F-hom M h)) ⟩ - ((F-hom M (f ⋆ g) ⋆ F-hom M (F-hom M h)) ⋆ N-ob μ y') - ≡⟨ ⋆Assoc (F-hom M (f ⋆ g)) (F-hom M (F-hom M h)) (N-ob μ y') ⟩ - (F-hom M (f ⋆ g) ⋆ (F-hom M (F-hom M h) ⋆ N-ob μ y')) - ≡⟨ cong (F-hom M (f ⋆ g) ⋆_) (N-hom μ h) ⟩ - (F-hom M (f ⋆ g) ⋆ (N-ob μ y ⋆ F-hom M h)) - ≡⟨ sym (⋆Assoc (F-hom M (f ⋆ g)) (N-ob μ y) (F-hom M h)) ⟩ - ((F-hom M (f ⋆ g) ⋆ N-ob μ y) ⋆ F-hom M h) - ≡⟨ cong (_⋆ F-hom M h) (cong (_⋆ N-ob μ y) (F-seq M f g)) ⟩ - (((F-hom M f ⋆ F-hom M g) ⋆ N-ob μ y) ⋆ F-hom M h) - ≡⟨ cong (_⋆ F-hom M h) (⋆Assoc (F-hom M f) (F-hom M g) (N-ob μ y)) ⟩ - ((M .F-hom f ⋆ (F-hom M g ⋆ N-ob μ y)) ⋆ F-hom M h) ∎ + (F-hom M ((f ⋆ g) ⋆ F-hom M h) ⋆ N-ob μ y') + ≡⟨ cong (_⋆ N-ob μ y') (F-seq M (f ⋆ g) (F-hom M h)) ⟩ + ((F-hom M (f ⋆ g) ⋆ F-hom M (F-hom M h)) ⋆ N-ob μ y') + ≡⟨ ⋆Assoc (F-hom M (f ⋆ g)) (F-hom M (F-hom M h)) (N-ob μ y') ⟩ + (F-hom M (f ⋆ g) ⋆ (F-hom M (F-hom M h) ⋆ N-ob μ y')) + ≡⟨ cong (F-hom M (f ⋆ g) ⋆_) (N-hom μ h) ⟩ + (F-hom M (f ⋆ g) ⋆ (N-ob μ y ⋆ F-hom M h)) + ≡⟨ sym (⋆Assoc (F-hom M (f ⋆ g)) (N-ob μ y) (F-hom M h)) ⟩ + ((F-hom M (f ⋆ g) ⋆ N-ob μ y) ⋆ F-hom M h) + ≡⟨ cong (_⋆ F-hom M h) (cong (_⋆ N-ob μ y) (F-seq M f g)) ⟩ + (((F-hom M f ⋆ F-hom M g) ⋆ N-ob μ y) ⋆ F-hom M h) + ≡⟨ cong (_⋆ F-hom M h) (⋆Assoc (F-hom M f) (F-hom M g) (N-ob μ y)) ⟩ + ((M .F-hom f ⋆ (F-hom M g ⋆ N-ob μ y)) ⋆ F-hom M h) ∎ -- Define comonads as monads on the opposite category? module _ (C : Category ℓ ℓ') where Monad : Type (ℓ-max ℓ ℓ') - Monad = Σ[ M ∈ Functor C C ] IsMonad M + Monad = Σ[ M ∈ Functor C C ] IsMonad M module _ {C : Category ℓ ℓ'} (monadM monadN : Monad C) (ν : NatTrans (fst monadM) (fst monadN)) where private - M N : Functor C C + M N : Functor C C M = fst monadM N = fst monadN module M = IsMonad (snd monadM) @@ -73,8 +73,8 @@ open IsMonadHom isProp-IsMonadHom : isProp (IsMonadHom) - N-η (isProp-IsMonadHom (proveMonadHom N-η1 N-μ1) (proveMonadHom N-η2 N-μ2) i) = isSetNatTrans _ _ N-η1 N-η2 i - N-μ (isProp-IsMonadHom (proveMonadHom N-η1 N-μ1) (proveMonadHom N-η2 N-μ2) i) = isSetNatTrans _ _ N-μ1 N-μ2 i + N-η (isProp-IsMonadHom (proveMonadHom N-η1 N-μ1) (proveMonadHom N-η2 N-μ2) i) = isSetNatTrans _ _ N-η1 N-η2 i + N-μ (isProp-IsMonadHom (proveMonadHom N-η1 N-μ1) (proveMonadHom N-η2 N-μ2) i) = isSetNatTrans _ _ N-μ1 N-μ2 i module _ {C : Category ℓ ℓ'} (monadM monadN : Monad C) where MonadHom : Type (ℓ-max ℓ ℓ') diff --git a/Cubical.Categories.Monoidal.Base.html b/Cubical.Categories.Monoidal.Base.html index 97d615826b..d31ddeca56 100644 --- a/Cubical.Categories.Monoidal.Base.html +++ b/Cubical.Categories.Monoidal.Base.html @@ -17,18 +17,18 @@ record TensorStr : Type (ℓ-max ℓ ℓ') where field - ─⊗─ : Functor (C ×C C) C + ─⊗─ : Functor (C ×C C) C unit : ob - open Functor + open Functor -- Useful tensor product notation _⊗_ : ob → ob → ob - x ⊗ y = ─⊗─ .F-ob (x , y) + x ⊗ y = ─⊗─ .F-ob (x , y) _⊗ₕ_ : ∀ {x y z w} → Hom[ x , y ] → Hom[ z , w ] → Hom[ x ⊗ z , y ⊗ w ] - f ⊗ₕ g = ─⊗─ .F-hom (f , g) + f ⊗ₕ g = ─⊗─ .F-hom (f , g) record StrictMonStr : Type (ℓ-max ℓ ℓ') where @@ -52,15 +52,15 @@ private -- Private names to make the axioms below look nice - x⊗[y⊗z] : Functor (C ×C C ×C C) C - x⊗[y⊗z] = ─⊗─ ∘F (𝟙⟨ C ⟩ ×F ─⊗─) + x⊗[y⊗z] : Functor (C ×C C ×C C) C + x⊗[y⊗z] = ─⊗─ ∘F (𝟙⟨ C ⟩ ×F ─⊗─) - [x⊗y]⊗z : Functor (C ×C C ×C C) C - [x⊗y]⊗z = ─⊗─ ∘F (─⊗─ ×F 𝟙⟨ C ⟩) ∘F (×C-assoc C C C) + [x⊗y]⊗z : Functor (C ×C C ×C C) C + [x⊗y]⊗z = ─⊗─ ∘F (─⊗─ ×F 𝟙⟨ C ⟩) ∘F (×C-assoc C C C) - x = 𝟙⟨ C ⟩ - 1⊗x = ─⊗─ ∘F (rinj C C unit) - x⊗1 = ─⊗─ ∘F (linj C C unit) + x = 𝟙⟨ C ⟩ + 1⊗x = ─⊗─ ∘F (rinj C C unit) + x⊗1 = ─⊗─ ∘F (linj C C unit) field -- "Axioms" - up to natural isomorphism diff --git a/Cubical.Categories.Monoidal.Strict.Monoid.html b/Cubical.Categories.Monoidal.Strict.Monoid.html index 16c18536b0..cc11ce91f5 100644 --- a/Cubical.Categories.Monoidal.Strict.Monoid.html +++ b/Cubical.Categories.Monoidal.Strict.Monoid.html @@ -49,7 +49,7 @@ open Category open Monoid open Monoid[_,_] -open Functor +open Functor monoidCategory : Category (ℓ-max ℓ ℓ') ℓ' ob monoidCategory = Monoid @@ -62,7 +62,7 @@ μ monA ≡⟨ sym (C.⋆IdL (μ monA)) ⟩ μ monA C.∘ id C.C - ≡⟨ cong (μ monA C.∘_) (sym (F-id C.─⊗─)) ⟩ + ≡⟨ cong (μ monA C.∘_) (sym (F-id C.─⊗─)) ⟩ μ monA C.∘ (C.id C.⊗ₕ C.id) ∎ carrierHom (_⋆_ monoidCategory f g) = carrierHom g C.∘ carrierHom f nat-η (_⋆_ monoidCategory {monA} {monB} {monC} f g) = @@ -85,7 +85,7 @@ (μ monC C.∘ (carrierHom g C.⊗ₕ carrierHom g)) C.∘ (carrierHom f C.⊗ₕ carrierHom f) ≡⟨ sym (C.⋆Assoc (carrierHom f C.⊗ₕ carrierHom f) (carrierHom g C.⊗ₕ carrierHom g) (μ monC)) ⟩ μ monC C.∘ ((carrierHom g C.⊗ₕ carrierHom g) C.∘ (carrierHom f C.⊗ₕ carrierHom f)) - ≡⟨ cong (μ monC C.∘_) (sym (F-seq C.─⊗─ (carrierHom f , carrierHom f) (carrierHom g , carrierHom g))) ⟩ + ≡⟨ cong (μ monC C.∘_) (sym (F-seq C.─⊗─ (carrierHom f , carrierHom f) (carrierHom g , carrierHom g))) ⟩ μ monC C.∘ ((carrierHom g C.∘ carrierHom f) C.⊗ₕ (carrierHom g C.∘ carrierHom f)) ∎ ⋆IdL monoidCategory {monA} f = monoidHomExt _ _ (C.⋆IdL (carrierHom f)) ⋆IdR monoidCategory {monA} f = monoidHomExt _ _ (C.⋆IdR (carrierHom f)) diff --git a/Cubical.Categories.NaturalTransformation.Base.html b/Cubical.Categories.NaturalTransformation.Base.html index 8f9c9aaad1..fa868bd427 100644 --- a/Cubical.Categories.NaturalTransformation.Base.html +++ b/Cubical.Categories.NaturalTransformation.Base.html @@ -26,16 +26,16 @@ f ⋆ᴰ g = f ⋆⟨ D ⟩ g open Category - open Functor + open Functor -- type aliases because it gets tedious typing it out all the time - N-ob-Type : (F G : Functor C D) → Type _ - N-ob-Type F G = (x : C .ob) → D [(F .F-ob x) , (G .F-ob x)] + N-ob-Type : (F G : Functor C D) → Type _ + N-ob-Type F G = (x : C .ob) → D [(F .F-ob x) , (G .F-ob x)] - N-hom-Type : (F G : Functor C D) → N-ob-Type F G → Type _ - N-hom-Type F G ϕ = {x y : C .ob} (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (ϕ y) ≡ (ϕ x) ⋆ᴰ (G .F-hom f) + N-hom-Type : (F G : Functor C D) → N-ob-Type F G → Type _ + N-hom-Type F G ϕ = {x y : C .ob} (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (ϕ y) ≡ (ϕ x) ⋆ᴰ (G .F-hom f) - record NatTrans (F G : Functor C D) : Type (ℓ-max (ℓ-max ℓC ℓC') ℓD') where + record NatTrans (F G : Functor C D) : Type (ℓ-max (ℓ-max ℓC ℓC') ℓD') where constructor natTrans field -- components of the natural transformation @@ -43,7 +43,7 @@ -- naturality condition N-hom : N-hom-Type F G N-ob - record NatIso (F G : Functor C D): Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where + record NatIso (F G : Functor C D): Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) where field trans : NatTrans F G open NatTrans trans @@ -55,103 +55,103 @@ -- the three other commuting squares sqRL : ∀ {x y : C .ob} {f : C [ x , y ]} - → F ⟪ f ⟫ ≡ (N-ob x) ⋆ᴰ G ⟪ f ⟫ ⋆ᴰ (nIso y) .inv + → F ⟪ f ⟫ ≡ (N-ob x) ⋆ᴰ G ⟪ f ⟫ ⋆ᴰ (nIso y) .inv sqRL {x} {y} {f} = invMoveR (isIso→areInv (nIso y)) (N-hom f) sqLL : ∀ {x y : C .ob} {f : C [ x , y ]} - → G ⟪ f ⟫ ⋆ᴰ (nIso y) .inv ≡ (nIso x) .inv ⋆ᴰ F ⟪ f ⟫ + → G ⟪ f ⟫ ⋆ᴰ (nIso y) .inv ≡ (nIso x) .inv ⋆ᴰ F ⟪ f ⟫ sqLL {x} {y} {f} = invMoveL (isIso→areInv (nIso x)) (sym sqRL') where - sqRL' : F ⟪ f ⟫ ≡ (N-ob x) ⋆ᴰ ( G ⟪ f ⟫ ⋆ᴰ (nIso y) .inv ) + sqRL' : F ⟪ f ⟫ ≡ (N-ob x) ⋆ᴰ ( G ⟪ f ⟫ ⋆ᴰ (nIso y) .inv ) sqRL' = sqRL ∙ (D .⋆Assoc _ _ _) sqLR : ∀ {x y : C .ob} {f : C [ x , y ]} - → G ⟪ f ⟫ ≡ (nIso x) .inv ⋆ᴰ F ⟪ f ⟫ ⋆ᴰ (N-ob y) + → G ⟪ f ⟫ ≡ (nIso x) .inv ⋆ᴰ F ⟪ f ⟫ ⋆ᴰ (N-ob y) sqLR {x} {y} {f} = invMoveR (symAreInv (isIso→areInv (nIso y))) sqLL open NatTrans open NatIso infix 10 _⇒_ - _⇒_ : Functor C D → Functor C D → Type (ℓ-max (ℓ-max ℓC ℓC') ℓD') + _⇒_ : Functor C D → Functor C D → Type (ℓ-max (ℓ-max ℓC ℓC') ℓD') _⇒_ = NatTrans infix 9 _≅ᶜ_ -- c superscript to indicate that this is in the context of categories - _≅ᶜ_ : Functor C D → Functor C D → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) + _≅ᶜ_ : Functor C D → Functor C D → Type (ℓ-max (ℓ-max ℓC ℓC') (ℓ-max ℓD ℓD')) _≅ᶜ_ = NatIso -- component of a natural transformation infix 30 _⟦_⟧ - _⟦_⟧ : ∀ {F G : Functor C D} → F ⇒ G → (x : C .ob) → D [ F .F-ob x , G .F-ob x ] + _⟦_⟧ : ∀ {F G : Functor C D} → F ⇒ G → (x : C .ob) → D [ F .F-ob x , G .F-ob x ] _⟦_⟧ = N-ob - idTrans : (F : Functor C D) → NatTrans F F + idTrans : (F : Functor C D) → NatTrans F F idTrans F .N-ob x = D .id idTrans F .N-hom f = - (F .F-hom f) ⋆ᴰ (idTrans F .N-ob _) + (F .F-hom f) ⋆ᴰ (idTrans F .N-ob _) ≡⟨ D .⋆IdR _ ⟩ - F .F-hom f + F .F-hom f ≡⟨ sym (D .⋆IdL _) ⟩ - (D .id) ⋆ᴰ (F .F-hom f) + (D .id) ⋆ᴰ (F .F-hom f) ∎ - 1[_] : (F : Functor C D) → NatTrans F F + 1[_] : (F : Functor C D) → NatTrans F F 1[_] = idTrans - idNatIso : (F : Functor C D) → NatIso F F + idNatIso : (F : Functor C D) → NatIso F F idNatIso F .trans = idTrans F idNatIso F .nIso _ = idCatIso .snd -- Natural isomorphism induced by path of functors - pathToNatTrans : {F G : Functor C D} → F ≡ G → NatTrans F G - pathToNatTrans p .N-ob x = pathToIso {C = D} (λ i → p i .F-ob x) .fst + pathToNatTrans : {F G : Functor C D} → F ≡ G → NatTrans F G + pathToNatTrans p .N-ob x = pathToIso {C = D} (λ i → p i .F-ob x) .fst pathToNatTrans {F = F} {G = G} p .N-hom {x = x} {y = y} f = - pathToIso-Comm {C = D} _ _ _ _ (λ i → p i .F-hom f) + pathToIso-Comm {C = D} _ _ _ _ (λ i → p i .F-hom f) - pathToNatIso : {F G : Functor C D} → F ≡ G → NatIso F G + pathToNatIso : {F G : Functor C D} → F ≡ G → NatIso F G pathToNatIso p .trans = pathToNatTrans p pathToNatIso p .nIso x = pathToIso {C = D} _ .snd -- vertical sequencing - seqTrans : {F G H : Functor C D} (α : NatTrans F G) (β : NatTrans G H) → NatTrans F H + seqTrans : {F G H : Functor C D} (α : NatTrans F G) (β : NatTrans G H) → NatTrans F H seqTrans α β .N-ob x = (α .N-ob x) ⋆ᴰ (β .N-ob x) seqTrans {F} {G} {H} α β .N-hom f = - (F .F-hom f) ⋆ᴰ ((α .N-ob _) ⋆ᴰ (β .N-ob _)) + (F .F-hom f) ⋆ᴰ ((α .N-ob _) ⋆ᴰ (β .N-ob _)) ≡⟨ sym (D .⋆Assoc _ _ _) ⟩ - ((F .F-hom f) ⋆ᴰ (α .N-ob _)) ⋆ᴰ (β .N-ob _) + ((F .F-hom f) ⋆ᴰ (α .N-ob _)) ⋆ᴰ (β .N-ob _) ≡[ i ]⟨ (α .N-hom f i) ⋆ᴰ (β .N-ob _) ⟩ - ((α .N-ob _) ⋆ᴰ (G .F-hom f)) ⋆ᴰ (β .N-ob _) + ((α .N-ob _) ⋆ᴰ (G .F-hom f)) ⋆ᴰ (β .N-ob _) ≡⟨ D .⋆Assoc _ _ _ ⟩ - (α .N-ob _) ⋆ᴰ ((G .F-hom f) ⋆ᴰ (β .N-ob _)) + (α .N-ob _) ⋆ᴰ ((G .F-hom f) ⋆ᴰ (β .N-ob _)) ≡[ i ]⟨ (α .N-ob _) ⋆ᴰ (β .N-hom f i) ⟩ - (α .N-ob _) ⋆ᴰ ((β .N-ob _) ⋆ᴰ (H .F-hom f)) + (α .N-ob _) ⋆ᴰ ((β .N-ob _) ⋆ᴰ (H .F-hom f)) ≡⟨ sym (D .⋆Assoc _ _ _) ⟩ - ((α .N-ob _) ⋆ᴰ (β .N-ob _)) ⋆ᴰ (H .F-hom f) + ((α .N-ob _) ⋆ᴰ (β .N-ob _)) ⋆ᴰ (H .F-hom f) ∎ - compTrans : {F G H : Functor C D} (β : NatTrans G H) (α : NatTrans F G) → NatTrans F H + compTrans : {F G H : Functor C D} (β : NatTrans G H) (α : NatTrans F G) → NatTrans F H compTrans β α = seqTrans α β infixl 8 _●ᵛ_ - _●ᵛ_ : {F G H : Functor C D} → NatTrans F G → NatTrans G H → NatTrans F H + _●ᵛ_ : {F G H : Functor C D} → NatTrans F G → NatTrans G H → NatTrans F H _●ᵛ_ = seqTrans -- vertically sequence natural transformations whose -- common functor is not definitional equal - seqTransP : {F G G' H : Functor C D} (p : G ≡ G') + seqTransP : {F G G' H : Functor C D} (p : G ≡ G') → (α : NatTrans F G) (β : NatTrans G' H) → NatTrans F H seqTransP {F} {G} {G'} {H} p α β .N-ob x -- sequence morphisms with non-judgementally equal (co)domain = seqP {C = D} {p = Gx≡G'x} (α ⟦ x ⟧) (β ⟦ x ⟧) where - Gx≡G'x : ∀ {x} → G ⟅ x ⟆ ≡ G' ⟅ x ⟆ - Gx≡G'x {x} i = F-ob (p i) x + Gx≡G'x : ∀ {x} → G ⟅ x ⟆ ≡ G' ⟅ x ⟆ + Gx≡G'x {x} i = F-ob (p i) x seqTransP {F} {G} {G'} {H} p α β .N-hom {x = x} {y} f -- compose the two commuting squares -- 1. α's commuting square @@ -159,74 +159,74 @@ = compSq {C = D} (α .N-hom f) βSq where -- functor equality implies equality of actions on objects and morphisms - Gx≡G'x : G ⟅ x ⟆ ≡ G' ⟅ x ⟆ - Gx≡G'x i = F-ob (p i) x + Gx≡G'x : G ⟅ x ⟆ ≡ G' ⟅ x ⟆ + Gx≡G'x i = F-ob (p i) x - Gy≡G'y : G ⟅ y ⟆ ≡ G' ⟅ y ⟆ - Gy≡G'y i = F-ob (p i) y + Gy≡G'y : G ⟅ y ⟆ ≡ G' ⟅ y ⟆ + Gy≡G'y i = F-ob (p i) y - Gf≡G'f : PathP (λ i → D [ Gx≡G'x i , Gy≡G'y i ]) (G ⟪ f ⟫) (G' ⟪ f ⟫) - Gf≡G'f i = p i ⟪ f ⟫ + Gf≡G'f : PathP (λ i → D [ Gx≡G'x i , Gy≡G'y i ]) (G ⟪ f ⟫) (G' ⟪ f ⟫) + Gf≡G'f i = p i ⟪ f ⟫ -- components of β extended out to Gx and Gy respectively - βx' = subst (λ a → D [ a , H ⟅ x ⟆ ]) (sym Gx≡G'x) (β ⟦ x ⟧) - βy' = subst (λ a → D [ a , H ⟅ y ⟆ ]) (sym Gy≡G'y) (β ⟦ y ⟧) + βx' = subst (λ a → D [ a , H ⟅ x ⟆ ]) (sym Gx≡G'x) (β ⟦ x ⟧) + βy' = subst (λ a → D [ a , H ⟅ y ⟆ ]) (sym Gy≡G'y) (β ⟦ y ⟧) -- extensions are equal to originals - βy'≡βy : PathP (λ i → D [ Gy≡G'y i , H ⟅ y ⟆ ]) βy' (β ⟦ y ⟧) - βy'≡βy = symP (toPathP {A = λ i → D [ Gy≡G'y (~ i) , H ⟅ y ⟆ ]} refl) + βy'≡βy : PathP (λ i → D [ Gy≡G'y i , H ⟅ y ⟆ ]) βy' (β ⟦ y ⟧) + βy'≡βy = symP (toPathP {A = λ i → D [ Gy≡G'y (~ i) , H ⟅ y ⟆ ]} refl) - βx≡βx' : PathP (λ i → D [ Gx≡G'x (~ i) , H ⟅ x ⟆ ]) (β ⟦ x ⟧) βx' + βx≡βx' : PathP (λ i → D [ Gx≡G'x (~ i) , H ⟅ x ⟆ ]) (β ⟦ x ⟧) βx' βx≡βx' = toPathP refl -- left wall of square - left : PathP (λ i → D [ Gx≡G'x i , H ⟅ y ⟆ ]) (G ⟪ f ⟫ ⋆⟨ D ⟩ βy') (G' ⟪ f ⟫ ⋆⟨ D ⟩ β ⟦ y ⟧) + left : PathP (λ i → D [ Gx≡G'x i , H ⟅ y ⟆ ]) (G ⟪ f ⟫ ⋆⟨ D ⟩ βy') (G' ⟪ f ⟫ ⋆⟨ D ⟩ β ⟦ y ⟧) left i = Gf≡G'f i ⋆⟨ D ⟩ βy'≡βy i -- right wall of square - right : PathP (λ i → D [ Gx≡G'x (~ i) , H ⟅ y ⟆ ]) (β ⟦ x ⟧ ⋆⟨ D ⟩ H ⟪ f ⟫) (βx' ⋆⟨ D ⟩ H ⟪ f ⟫) - right i = βx≡βx' i ⋆⟨ D ⟩ refl {x = H ⟪ f ⟫} i + right : PathP (λ i → D [ Gx≡G'x (~ i) , H ⟅ y ⟆ ]) (β ⟦ x ⟧ ⋆⟨ D ⟩ H ⟪ f ⟫) (βx' ⋆⟨ D ⟩ H ⟪ f ⟫) + right i = βx≡βx' i ⋆⟨ D ⟩ refl {x = H ⟪ f ⟫} i -- putting it all together - βSq : G ⟪ f ⟫ ⋆⟨ D ⟩ βy' ≡ βx' ⋆⟨ D ⟩ H ⟪ f ⟫ - βSq i = comp (λ k → D [ Gx≡G'x (~ k) , H ⟅ y ⟆ ]) + βSq : G ⟪ f ⟫ ⋆⟨ D ⟩ βy' ≡ βx' ⋆⟨ D ⟩ H ⟪ f ⟫ + βSq i = comp (λ k → D [ Gx≡G'x (~ k) , H ⟅ y ⟆ ]) (λ j → λ { (i = i0) → left (~ j) ; (i = i1) → right j }) (β .N-hom f i) - module _ {F G : Functor C D} {α β : NatTrans F G} where + module _ {F G : Functor C D} {α β : NatTrans F G} where open Category - open Functor + open Functor open NatTrans makeNatTransPath : α .N-ob ≡ β .N-ob → α ≡ β makeNatTransPath p i .N-ob = p i makeNatTransPath p i .N-hom f = rem i where - rem : PathP (λ i → (F .F-hom f) ⋆ᴰ (p i _) ≡ (p i _) ⋆ᴰ (G .F-hom f)) + rem : PathP (λ i → (F .F-hom f) ⋆ᴰ (p i _) ≡ (p i _) ⋆ᴰ (G .F-hom f)) (α .N-hom f) (β .N-hom f) rem = toPathP (D .isSetHom _ _ _ _) -- `constructor` for path of natural isomorphisms - NatIso≡ : {F G : Functor C D}{f g : NatIso F G} → f .trans .N-ob ≡ g .trans .N-ob → f ≡ g + NatIso≡ : {F G : Functor C D}{f g : NatIso F G} → f .trans .N-ob ≡ g .trans .N-ob → f ≡ g NatIso≡ {f = f} {g} p i .trans = makeNatTransPath {α = f .trans} {β = g .trans} p i NatIso≡ {f = f} {g} p i .nIso x = isProp→PathP (λ i → isPropIsIso (NatIso≡ {f = f} {g} p i .trans .N-ob x)) (f .nIso _) (g .nIso _) i - module _ {F F' G G' : Functor C D} {α : NatTrans F G} {β : NatTrans F' G'} where + module _ {F F' G G' : Functor C D} {α : NatTrans F G} {β : NatTrans F' G'} where open Category - open Functor + open Functor open NatTrans makeNatTransPathP : ∀ (p : F ≡ F') (q : G ≡ G') - → PathP (λ i → (x : C .ob) → D [ (p i) .F-ob x , (q i) .F-ob x ]) + → PathP (λ i → (x : C .ob) → D [ (p i) .F-ob x , (q i) .F-ob x ]) (α .N-ob) (β .N-ob) → PathP (λ i → NatTrans (p i) (q i)) α β makeNatTransPathP p q P i .N-ob = P i makeNatTransPathP p q P i .N-hom f = rem i where - rem : PathP (λ i → ((p i) .F-hom f) ⋆ᴰ (P i _) ≡ (P i _) ⋆ᴰ ((q i) .F-hom f)) + rem : PathP (λ i → ((p i) .F-hom f) ⋆ᴰ (P i _) ≡ (P i _) ⋆ᴰ ((q i) .F-hom f)) (α .N-hom f) (β .N-hom f) rem = toPathP (D .isSetHom _ _ _ _) @@ -234,18 +234,18 @@ open NatTrans -- whiskering -- αF - _∘ˡ_ : ∀ {G H : Functor C D} (α : NatTrans G H) → (F : Functor B C) - → NatTrans (G ∘F F) (H ∘F F) - (_∘ˡ_ {G} {H} α F) .N-ob x = α ⟦ F ⟅ x ⟆ ⟧ + _∘ˡ_ : ∀ {G H : Functor C D} (α : NatTrans G H) → (F : Functor B C) + → NatTrans (G ∘F F) (H ∘F F) + (_∘ˡ_ {G} {H} α F) .N-ob x = α ⟦ F ⟅ x ⟆ ⟧ (_∘ˡ_ {G} {H} α F) .N-hom f = (α .N-hom) _ -- Kβ - _∘ʳ_ : ∀ (K : Functor C D) → {G H : Functor B C} (β : NatTrans G H) - → NatTrans (K ∘F G) (K ∘F H) - (_∘ʳ_ K {G} {H} β) .N-ob x = K ⟪ β ⟦ x ⟧ ⟫ + _∘ʳ_ : ∀ (K : Functor C D) → {G H : Functor B C} (β : NatTrans G H) + → NatTrans (K ∘F G) (K ∘F H) + (_∘ʳ_ K {G} {H} β) .N-ob x = K ⟪ β ⟦ x ⟧ ⟫ (_∘ʳ_ K {G} {H} β) .N-hom f = preserveCommF {C = C} {D = D} {K} (β .N-hom f) - whiskerTrans : {F F' : Functor B C} {G G' : Functor C D} (β : NatTrans G G') (α : NatTrans F F') - → NatTrans (G ∘F F) (G' ∘F F') + whiskerTrans : {F F' : Functor B C} {G G' : Functor C D} (β : NatTrans G G') (α : NatTrans F F') + → NatTrans (G ∘F F) (G' ∘F F') whiskerTrans {F}{F'}{G}{G'} β α = compTrans (β ∘ˡ F') (G ∘ʳ α) \ No newline at end of file diff --git a/Cubical.Categories.NaturalTransformation.Properties.html b/Cubical.Categories.NaturalTransformation.Properties.html index 3a165f8d02..cd80b5ba8a 100644 --- a/Cubical.Categories.NaturalTransformation.Properties.html +++ b/Cubical.Categories.NaturalTransformation.Properties.html @@ -21,156 +21,177 @@ private variable ℓB ℓB' ℓC ℓC' ℓD ℓD' ℓE ℓE' : Level + C : Category ℓC ℓC' + D : Category ℓD ℓD' + F F' : Functor C D + +open isIsoC +open NatIso +open NatTrans +open Category +open Functor +open Iso + +module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} where + private + _⋆ᴰ_ : ∀ {x y z} (f : D [ x , y ]) (g : D [ y , z ]) → D [ x , z ] + f ⋆ᴰ g = f ⋆⟨ D ⟩ g + + -- natural isomorphism is symmetric + symNatIso : ∀ {F G : Functor C D} + → F ≅ᶜ G + → G ≅ᶜ F + symNatIso η .trans .N-ob x = η .nIso x .inv + symNatIso η .trans .N-hom _ = sqLL η + symNatIso η .nIso x .inv = η .trans .N-ob x + symNatIso η .nIso x .sec = η .nIso x .ret + symNatIso η .nIso x .ret = η .nIso x .sec + + -- Properties + + -- path helpers + module NatTransP where + + module _ {F G : Functor C D} where + + -- same as Sigma version + NatTransΣ : Type (ℓ-max (ℓ-max ℓC ℓC') ℓD') + NatTransΣ = Σ[ ob ∈ ((x : C .ob) → D [(F .F-ob x) , (G .F-ob x)]) ] + ({x y : _ } (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (ob y) ≡ (ob x) ⋆ᴰ (G .F-hom f)) + + NatTransIsoΣ : Iso (NatTrans F G) NatTransΣ + NatTransIsoΣ .fun (natTrans N-ob N-hom) = N-ob , N-hom + NatTransIsoΣ .inv (N-ob , N-hom) = (natTrans N-ob N-hom) + NatTransIsoΣ .rightInv _ = refl + NatTransIsoΣ .leftInv _ = refl + + NatTrans≡Σ : NatTrans F G ≡ NatTransΣ + NatTrans≡Σ = isoToPath NatTransIsoΣ + + -- introducing paths + NatTrans-≡-intro : ∀ {αo βo : N-ob-Type F G} + {αh : N-hom-Type F G αo} + {βh : N-hom-Type F G βo} + → (p : αo ≡ βo) + → PathP (λ i → ({x y : C .ob} (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ (G .F-hom f))) αh βh + → natTrans {F = F} {G} αo αh ≡ natTrans βo βh + NatTrans-≡-intro p q i = natTrans (p i) (q i) + + module _ {F G : Functor C D} {α β : NatTrans F G} where + open Iso + private + αOb = α .N-ob + βOb = β .N-ob + αHom = α .N-hom + βHom = β .N-hom + -- path between natural transformations is the same as a pair of paths (between ob and hom) + NTPathIsoPathΣ : Iso (α ≡ β) + (Σ[ p ∈ (αOb ≡ βOb) ] + (PathP (λ i → ({x y : _} (f : _) → F ⟪ f ⟫ ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ G ⟪ f ⟫)) + αHom + βHom)) + NTPathIsoPathΣ .fun p = (λ i → p i .N-ob) , (λ i → p i .N-hom) + NTPathIsoPathΣ .inv (po , ph) i = record { N-ob = po i ; N-hom = ph i } + NTPathIsoPathΣ .rightInv pσ = refl + NTPathIsoPathΣ .leftInv p = refl + + NTPath≃PathΣ = isoToEquiv NTPathIsoPathΣ + + NTPath≡PathΣ = ua NTPath≃PathΣ + + module _ where + open NatTransP + + isSetNatTrans : {F G : Functor C D} → isSet (NatTrans F G) + isSetNatTrans = + isSetRetract (fun NatTransIsoΣ) (inv NatTransIsoΣ) (leftInv NatTransIsoΣ) + (isSetΣSndProp (isSetΠ (λ _ → isSetHom D)) + (λ _ → isPropImplicitΠ2 (λ _ _ → isPropΠ (λ _ → isSetHom D _ _)))) + + +-- Natural isomorphism is path when the target category is univalent. + +module _ + (isUnivD : isUnivalent D) + {F G : Functor C D} where + + open isUnivalent isUnivD + + NatIsoToPath : NatIso F G → F ≡ G + NatIsoToPath niso = + Functor≡ (λ x → CatIsoToPath (_ , niso .nIso x)) + (λ f → isoToPath-Square isUnivD _ _ _ _ (niso .trans .N-hom f)) + + NatIso→Path→NatIso : (niso : NatIso F G) → pathToNatIso (NatIsoToPath niso) ≡ niso + NatIso→Path→NatIso niso = NatIso≡ (λ i x → secEq (univEquiv _ _) (_ , niso .nIso x) i .fst) + + Path→NatIso→Path : (p : F ≡ G) → NatIsoToPath (pathToNatIso p) ≡ p + Path→NatIso→Path p = FunctorPath≡ (λ i j x → retEq (univEquiv _ _) (λ i → p i .F-ob x) i j) + + Iso-Path-NatIso : Iso (F ≡ G) (NatIso F G) + Iso-Path-NatIso = iso pathToNatIso NatIsoToPath NatIso→Path→NatIso Path→NatIso→Path + + Path≃NatIso : (F ≡ G) ≃ NatIso F G + Path≃NatIso = isoToEquiv Iso-Path-NatIso +module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} where + seqNatIso : {F G H : Functor C D} → NatIso F G → NatIso G H → NatIso F H + seqNatIso ı ı' .trans = seqTrans (ı .trans) (ı' .trans) + seqNatIso ı ı' .nIso x .inv = ı' .nIso x .inv ⋆⟨ D ⟩ ı .nIso x .inv + seqNatIso ı ı' .nIso x .sec = + D .⋆Assoc _ _ _ + ∙ cong (_⋆_ D (ı' .nIso x .inv)) + (sym (D .⋆Assoc _ _ _) + ∙ cong (D ∘ ı' .trans .N-ob x) (ı .nIso x .sec) + ∙ D .⋆IdL (ı' .trans .N-ob x)) + ∙ ı' .nIso x .sec + seqNatIso ı ı' .nIso x .ret = + (sym (D .⋆Assoc _ _ _)) + ∙ cong (_∘_ D (ı .nIso x .inv)) + (D .⋆Assoc _ _ _ + ∙ cong (D ⋆ ı .trans .N-ob x) (ı' .nIso x .ret) + ∙ D .⋆IdR (ı .trans .N-ob x)) + ∙ ı .nIso x .ret + + CAT⋆IdR : {F : Functor C D} → NatIso (Id ∘F F) F + CAT⋆IdR {F} .trans .N-ob = idTrans F .N-ob + CAT⋆IdR {F} .trans .N-hom = idTrans F .N-hom + CAT⋆IdR {F} .nIso = idNatIso F .nIso + +module _ {B : Category ℓB ℓB'}{C : Category ℓC ℓC'}{D : Category ℓD ℓD'} where + _∘ʳi_ : ∀ (K : Functor C D) → {G H : Functor B C} (β : NatIso G H) + → NatIso (K ∘F G) (K ∘F H) + _∘ʳi_ K β .trans = K ∘ʳ β .trans + _∘ʳi_ K β .nIso x = preserveIsosF {F = K} (β .trans .N-ob _ , β .nIso x) .snd + + open Functor + _∘ˡi_ : ∀ (K : Functor B C) → {G H : Functor C D} (β : NatIso G H) + → NatIso (G ∘F K) (H ∘F K) + _∘ˡi_ K β .trans = β .trans ∘ˡ K + _∘ˡi_ K β .nIso b = β .nIso (K ⟅ b ⟆) + + CAT⋆Assoc : {E : Category ℓE ℓE'} + (F : Functor B C)(G : Functor C D)(H : Functor D E) + → NatIso (H ∘F (G ∘F F)) ((H ∘F G) ∘F F) + CAT⋆Assoc F G H .trans .N-ob = idTrans ((H ∘F G) ∘F F) .N-ob + CAT⋆Assoc F G H .trans .N-hom = idTrans ((H ∘F G) ∘F F) .N-hom + CAT⋆Assoc F G H .nIso = idNatIso ((H ∘F G) ∘F F) .nIso + + + +⇒^opFiso : Iso (F ⇒ F') (_^opF {C = C} {D = D} F' ⇒ F ^opF ) +N-ob (fun ⇒^opFiso x) = N-ob x +N-hom (fun ⇒^opFiso x) f = sym (N-hom x f) +inv ⇒^opFiso = _ +rightInv ⇒^opFiso _ = refl +leftInv ⇒^opFiso _ = refl + +congNatIso^opFiso : Iso (F ≅ᶜ F') (_^opF {C = C} {D = D} F' ≅ᶜ F ^opF ) +trans (fun congNatIso^opFiso x) = Iso.fun ⇒^opFiso (trans x) +inv (nIso (fun congNatIso^opFiso x) x₁) = _ +sec (nIso (fun congNatIso^opFiso x) x₁) = ret (nIso x x₁) +ret (nIso (fun congNatIso^opFiso x) x₁) = sec (nIso x x₁) +inv congNatIso^opFiso = _ +rightInv congNatIso^opFiso _ = refl +leftInv congNatIso^opFiso _ = refl -open isIsoC -open NatIso -open NatTrans -open Category -open Functor -open Iso - -module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} where - private - _⋆ᴰ_ : ∀ {x y z} (f : D [ x , y ]) (g : D [ y , z ]) → D [ x , z ] - f ⋆ᴰ g = f ⋆⟨ D ⟩ g - - -- natural isomorphism is symmetric - symNatIso : ∀ {F G : Functor C D} - → F ≅ᶜ G - → G ≅ᶜ F - symNatIso η .trans .N-ob x = η .nIso x .inv - symNatIso η .trans .N-hom _ = sqLL η - symNatIso η .nIso x .inv = η .trans .N-ob x - symNatIso η .nIso x .sec = η .nIso x .ret - symNatIso η .nIso x .ret = η .nIso x .sec - - -- Properties - - -- path helpers - module NatTransP where - - module _ {F G : Functor C D} where - - -- same as Sigma version - NatTransΣ : Type (ℓ-max (ℓ-max ℓC ℓC') ℓD') - NatTransΣ = Σ[ ob ∈ ((x : C .ob) → D [(F .F-ob x) , (G .F-ob x)]) ] - ({x y : _ } (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (ob y) ≡ (ob x) ⋆ᴰ (G .F-hom f)) - - NatTransIsoΣ : Iso (NatTrans F G) NatTransΣ - NatTransIsoΣ .fun (natTrans N-ob N-hom) = N-ob , N-hom - NatTransIsoΣ .inv (N-ob , N-hom) = (natTrans N-ob N-hom) - NatTransIsoΣ .rightInv _ = refl - NatTransIsoΣ .leftInv _ = refl - - NatTrans≡Σ : NatTrans F G ≡ NatTransΣ - NatTrans≡Σ = isoToPath NatTransIsoΣ - - -- introducing paths - NatTrans-≡-intro : ∀ {αo βo : N-ob-Type F G} - {αh : N-hom-Type F G αo} - {βh : N-hom-Type F G βo} - → (p : αo ≡ βo) - → PathP (λ i → ({x y : C .ob} (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ (G .F-hom f))) αh βh - → natTrans {F = F} {G} αo αh ≡ natTrans βo βh - NatTrans-≡-intro p q i = natTrans (p i) (q i) - - module _ {F G : Functor C D} {α β : NatTrans F G} where - open Iso - private - αOb = α .N-ob - βOb = β .N-ob - αHom = α .N-hom - βHom = β .N-hom - -- path between natural transformations is the same as a pair of paths (between ob and hom) - NTPathIsoPathΣ : Iso (α ≡ β) - (Σ[ p ∈ (αOb ≡ βOb) ] - (PathP (λ i → ({x y : _} (f : _) → F ⟪ f ⟫ ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ G ⟪ f ⟫)) - αHom - βHom)) - NTPathIsoPathΣ .fun p = (λ i → p i .N-ob) , (λ i → p i .N-hom) - NTPathIsoPathΣ .inv (po , ph) i = record { N-ob = po i ; N-hom = ph i } - NTPathIsoPathΣ .rightInv pσ = refl - NTPathIsoPathΣ .leftInv p = refl - - NTPath≃PathΣ = isoToEquiv NTPathIsoPathΣ - - NTPath≡PathΣ = ua NTPath≃PathΣ - - module _ where - open NatTransP - - isSetNatTrans : {F G : Functor C D} → isSet (NatTrans F G) - isSetNatTrans = - isSetRetract (fun NatTransIsoΣ) (inv NatTransIsoΣ) (leftInv NatTransIsoΣ) - (isSetΣSndProp (isSetΠ (λ _ → isSetHom D)) - (λ _ → isPropImplicitΠ2 (λ _ _ → isPropΠ (λ _ → isSetHom D _ _)))) - - --- Natural isomorphism is path when the target category is univalent. - -module _ - {C : Category ℓC ℓC'} - {D : Category ℓD ℓD'}(isUnivD : isUnivalent D) - {F G : Functor C D} where - - open isUnivalent isUnivD - - NatIsoToPath : NatIso F G → F ≡ G - NatIsoToPath niso = - Functor≡ (λ x → CatIsoToPath (_ , niso .nIso x)) - (λ f → isoToPath-Square isUnivD _ _ _ _ (niso .trans .N-hom f)) - - NatIso→Path→NatIso : (niso : NatIso F G) → pathToNatIso (NatIsoToPath niso) ≡ niso - NatIso→Path→NatIso niso = NatIso≡ (λ i x → secEq (univEquiv _ _) (_ , niso .nIso x) i .fst) - - Path→NatIso→Path : (p : F ≡ G) → NatIsoToPath (pathToNatIso p) ≡ p - Path→NatIso→Path p = FunctorPath≡ (λ i j x → retEq (univEquiv _ _) (λ i → p i .F-ob x) i j) - - Iso-Path-NatIso : Iso (F ≡ G) (NatIso F G) - Iso-Path-NatIso = iso pathToNatIso NatIsoToPath NatIso→Path→NatIso Path→NatIso→Path - - Path≃NatIso : (F ≡ G) ≃ NatIso F G - Path≃NatIso = isoToEquiv Iso-Path-NatIso -module _ {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} where - seqNatIso : {F G H : Functor C D} → NatIso F G → NatIso G H → NatIso F H - seqNatIso ı ı' .trans = seqTrans (ı .trans) (ı' .trans) - seqNatIso ı ı' .nIso x .inv = ı' .nIso x .inv ⋆⟨ D ⟩ ı .nIso x .inv - seqNatIso ı ı' .nIso x .sec = - D .⋆Assoc _ _ _ - ∙ cong (_⋆_ D (ı' .nIso x .inv)) - (sym (D .⋆Assoc _ _ _) - ∙ cong (D ∘ ı' .trans .N-ob x) (ı .nIso x .sec) - ∙ D .⋆IdL (ı' .trans .N-ob x)) - ∙ ı' .nIso x .sec - seqNatIso ı ı' .nIso x .ret = - (sym (D .⋆Assoc _ _ _)) - ∙ cong (_∘_ D (ı .nIso x .inv)) - (D .⋆Assoc _ _ _ - ∙ cong (D ⋆ ı .trans .N-ob x) (ı' .nIso x .ret) - ∙ D .⋆IdR (ı .trans .N-ob x)) - ∙ ı .nIso x .ret - - CAT⋆IdR : {F : Functor C D} → NatIso (Id ∘F F) F - CAT⋆IdR {F} .trans .N-ob = idTrans F .N-ob - CAT⋆IdR {F} .trans .N-hom = idTrans F .N-hom - CAT⋆IdR {F} .nIso = idNatIso F .nIso - -module _ {B : Category ℓB ℓB'}{C : Category ℓC ℓC'}{D : Category ℓD ℓD'} where - _∘ʳi_ : ∀ (K : Functor C D) → {G H : Functor B C} (β : NatIso G H) - → NatIso (K ∘F G) (K ∘F H) - _∘ʳi_ K β .trans = K ∘ʳ β .trans - _∘ʳi_ K β .nIso x = preserveIsosF {F = K} (β .trans .N-ob _ , β .nIso x) .snd - - open Functor - _∘ˡi_ : ∀ (K : Functor B C) → {G H : Functor C D} (β : NatIso G H) - → NatIso (G ∘F K) (H ∘F K) - _∘ˡi_ K β .trans = β .trans ∘ˡ K - _∘ˡi_ K β .nIso b = β .nIso (K ⟅ b ⟆) - - CAT⋆Assoc : {E : Category ℓE ℓE'} - (F : Functor B C)(G : Functor C D)(H : Functor D E) - → NatIso (H ∘F (G ∘F F)) ((H ∘F G) ∘F F) - CAT⋆Assoc F G H .trans .N-ob = idTrans ((H ∘F G) ∘F F) .N-ob - CAT⋆Assoc F G H .trans .N-hom = idTrans ((H ∘F G) ∘F F) .N-hom - CAT⋆Assoc F G H .nIso = idNatIso ((H ∘F G) ∘F F) .nIso \ No newline at end of file diff --git a/Cubical.Categories.Presheaf.Base.html b/Cubical.Categories.Presheaf.Base.html index 446fcc07ea..7794e36bd1 100644 --- a/Cubical.Categories.Presheaf.Base.html +++ b/Cubical.Categories.Presheaf.Base.html @@ -14,7 +14,7 @@ ℓ ℓ' ℓS : Level Presheaf : Category ℓ ℓ' → (ℓS : Level) → Type (ℓ-max (ℓ-max ℓ ℓ') (ℓ-suc ℓS)) -Presheaf C ℓS = Functor (C ^op) (SET ℓS) +Presheaf C ℓS = Functor (C ^op) (SET ℓS) PresheafCategory : Category ℓ ℓ' → (ℓS : Level) → Category (ℓ-max (ℓ-max ℓ ℓ') (ℓ-suc ℓS)) @@ -26,24 +26,24 @@ isUnivalentPresheafCategory = isUnivalentFUNCTOR _ _ isUnivalentSET open Category -open Functor +open Functor action : ∀ (C : Category ℓ ℓ') → (P : Presheaf C ℓS) - → {a b : C .ob} → C [ a , b ] → fst (P ⟅ b ⟆) → fst (P ⟅ a ⟆) -action C P = P .F-hom + → {a b : C .ob} → C [ a , b ] → fst (P ⟅ b ⟆) → fst (P ⟅ a ⟆) +action C P = P .F-hom -- Convenient notation for naturality syntax action C P f ϕ = ϕ ∘ᴾ⟨ C , P ⟩ f ∘ᴾId : ∀ (C : Category ℓ ℓ') → (P : Presheaf C ℓS) → {a : C .ob} - → (ϕ : fst (P ⟅ a ⟆)) + → (ϕ : fst (P ⟅ a ⟆)) → ϕ ∘ᴾ⟨ C , P ⟩ C .id ≡ ϕ -∘ᴾId C P ϕ i = P .F-id i ϕ +∘ᴾId C P ϕ i = P .F-id i ϕ ∘ᴾAssoc : ∀ (C : Category ℓ ℓ') → (P : Presheaf C ℓS) → {a b c : C .ob} - → (ϕ : fst (P ⟅ c ⟆)) + → (ϕ : fst (P ⟅ c ⟆)) → (f : C [ b , c ]) → (g : C [ a , b ]) → ϕ ∘ᴾ⟨ C , P ⟩ (f ∘⟨ C ⟩ g) ≡ (ϕ ∘ᴾ⟨ C , P ⟩ f) ∘ᴾ⟨ C , P ⟩ g -∘ᴾAssoc C P ϕ f g i = P .F-seq f g i ϕ +∘ᴾAssoc C P ϕ f g i = P .F-seq f g i ϕ \ No newline at end of file diff --git a/Cubical.Categories.Presheaf.KanExtension.html b/Cubical.Categories.Presheaf.KanExtension.html index 7cd66ca697..0b258237b5 100644 --- a/Cubical.Categories.Presheaf.KanExtension.html +++ b/Cubical.Categories.Presheaf.KanExtension.html @@ -30,10 +30,10 @@ module Lan {ℓC ℓC' ℓD ℓD'} ℓS {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - (F : Functor C D) + (F : Functor C D) where - open Functor + open Functor open NatTrans private @@ -47,18 +47,18 @@ -} ℓ = ℓ-max (ℓ-max (ℓ-max ℓC ℓC') ℓD') ℓS - module _ (G : Functor (C ^op) (SET ℓ)) where + module _ (G : Functor (C ^op) (SET ℓ)) where -- Definition of the coend module _ (d : D.ob) where Raw : Type ℓ - Raw = Σ[ c ∈ C.ob ] Σ[ g ∈ D.Hom[ d , F ⟅ c ⟆ ] ] G .F-ob c .fst + Raw = Σ[ c ∈ C.ob ] Σ[ g ∈ D.Hom[ d , F ⟅ c ⟆ ] ] G .F-ob c .fst data _≈_ : (u v : Raw) → Type ℓ where - shift : {c c' : C.ob} (g : D.Hom[ d , F ⟅ c ⟆ ]) (f : C.Hom[ c , c' ]) (a : (G ⟅ c' ⟆) .fst) - → (c' , (g D.⋆ F ⟪ f ⟫) , a) ≈ (c , g , (G ⟪ f ⟫) a) + shift : {c c' : C.ob} (g : D.Hom[ d , F ⟅ c ⟆ ]) (f : C.Hom[ c , c' ]) (a : (G ⟅ c' ⟆) .fst) + → (c' , (g D.⋆ F ⟪ f ⟫) , a) ≈ (c , g , (G ⟪ f ⟫) a) Quo = Raw / _≈_ @@ -71,8 +71,8 @@ mapR h (shift/ g f a i) = hcomp (λ j → λ - { (i = i0) → [ _ , D.⋆Assoc h g (F ⟪ f ⟫) j , a ] - ; (i = i1) → [ _ , h D.⋆ g , (G ⟪ f ⟫) a ] + { (i = i0) → [ _ , D.⋆Assoc h g (F ⟪ f ⟫) j , a ] + ; (i = i1) → [ _ , h D.⋆ g , (G ⟪ f ⟫) a ] }) (shift/ (h D.⋆ g) f a i) mapR h (squash/ t u p q i j) = @@ -88,23 +88,23 @@ mapR∘ h' h = funExt (elimProp (λ _ → squash/ _ _) (λ (c , g , a) i → [ c , D.⋆Assoc h' h g i , a ])) - LanOb : Functor (C ^op) (SET ℓ) → Functor (D ^op) (SET _) - LanOb G .F-ob d .fst = Quo G d - LanOb G .F-ob d .snd = squash/ - LanOb G .F-hom = mapR G - LanOb G .F-id {d} = mapRId G d - LanOb G .F-seq h h' = mapR∘ G h' h + LanOb : Functor (C ^op) (SET ℓ) → Functor (D ^op) (SET _) + LanOb G .F-ob d .fst = Quo G d + LanOb G .F-ob d .snd = squash/ + LanOb G .F-hom = mapR G + LanOb G .F-id {d} = mapRId G d + LanOb G .F-seq h h' = mapR∘ G h' h -- Action of Quo on arrows in Cᵒᵖ → Set - module _ {G G' : Functor (C ^op) (SET ℓ)} (α : NatTrans G G') where + module _ {G G' : Functor (C ^op) (SET ℓ)} (α : NatTrans G G') where mapL : (d : D.ob) → Quo G d → Quo G' d mapL d [ c , g , a ] = [ c , g , α .N-ob c a ] mapL d (shift/ g f a i) = hcomp (λ j → λ - { (i = i0) → [ _ , (g D.⋆ F ⟪ f ⟫) , α .N-ob _ a ] + { (i = i0) → [ _ , (g D.⋆ F ⟪ f ⟫) , α .N-ob _ a ] ; (i = i1) → [ _ , g , funExt⁻ (α .N-hom f) a (~ j) ] }) (shift/ g f ((α ⟦ _ ⟧) a) i) @@ -115,87 +115,87 @@ → mapL d' ∘ mapR G h ≡ mapR G' h ∘ mapL d mapLR h = funExt (elimProp (λ _ → squash/ _ _) (λ _ → refl)) - mapLId : (G : Functor (C ^op) (SET ℓ)) + mapLId : (G : Functor (C ^op) (SET ℓ)) (d : D.ob) → mapL (idTrans G) d ≡ idfun (Quo G d) mapLId G d = funExt (elimProp (λ _ → squash/ _ _) (λ _ → refl)) - mapL∘ : {G G' G'' : Functor (C ^op) (SET ℓ)} + mapL∘ : {G G' G'' : Functor (C ^op) (SET ℓ)} (β : NatTrans G' G'') (α : NatTrans G G') (d : D.ob) → mapL (seqTrans α β) d ≡ mapL β d ∘ mapL α d mapL∘ β α d = funExt (elimProp (λ _ → squash/ _ _) (λ _ → refl)) - LanHom : {G G' : Functor (C ^op) (SET ℓ)} + LanHom : {G G' : Functor (C ^op) (SET ℓ)} → NatTrans G G' → NatTrans (LanOb G) (LanOb G') LanHom α .N-ob = mapL α LanHom α .N-hom = mapLR α -- Definition of the left Kan extension functor - Lan : Functor (FUNCTOR (C ^op) (SET ℓ)) (FUNCTOR (D ^op) (SET ℓ)) - Lan .F-ob = LanOb - Lan .F-hom = LanHom - Lan .F-id {G} = makeNatTransPath (funExt (mapLId G)) - Lan .F-seq α β = makeNatTransPath (funExt (mapL∘ β α)) + Lan : Functor (FUNCTOR (C ^op) (SET ℓ)) (FUNCTOR (D ^op) (SET ℓ)) + Lan .F-ob = LanOb + Lan .F-hom = LanHom + Lan .F-id {G} = makeNatTransPath (funExt (mapLId G)) + Lan .F-seq α β = makeNatTransPath (funExt (mapL∘ β α)) -- Adjunction between the left Kan extension and precomposition private - F* = precomposeF (SET ℓ) (F ^opF) + F* = precomposeF (SET ℓ) (F ^opF) - open UnitCounit + open UnitCounit - η : 𝟙⟨ FUNCTOR (C ^op) (SET ℓ) ⟩ ⇒ funcComp F* Lan + η : 𝟙⟨ FUNCTOR (C ^op) (SET ℓ) ⟩ ⇒ funcComp F* Lan η .N-ob G .N-ob c a = [ c , D.id , a ] η .N-ob G .N-hom {c'} {c} f = funExt λ a → - [ c , D.id , (G ⟪ f ⟫) a ] + [ c , D.id , (G ⟪ f ⟫) a ] ≡⟨ sym (shift/ D.id f a) ⟩ - [ c' , (D.id D.⋆ F ⟪ f ⟫) , a ] + [ c' , (D.id D.⋆ F ⟪ f ⟫) , a ] ≡[ i ]⟨ [ c' , lem i , a ] ⟩ - [ c' , (F ⟪ f ⟫ D.⋆ D.id) , a ] + [ c' , (F ⟪ f ⟫ D.⋆ D.id) , a ] ∎ where - lem : D.id D.⋆ F ⟪ f ⟫ ≡ F ⟪ f ⟫ D.⋆ D.id - lem = D.⋆IdL (F ⟪ f ⟫) ∙ sym (D.⋆IdR (F ⟪ f ⟫)) + lem : D.id D.⋆ F ⟪ f ⟫ ≡ F ⟪ f ⟫ D.⋆ D.id + lem = D.⋆IdL (F ⟪ f ⟫) ∙ sym (D.⋆IdR (F ⟪ f ⟫)) η .N-hom f = makeNatTransPath refl - ε : funcComp Lan F* ⇒ 𝟙⟨ FUNCTOR (D ^op) (SET ℓ) ⟩ + ε : funcComp Lan F* ⇒ 𝟙⟨ FUNCTOR (D ^op) (SET ℓ) ⟩ ε .N-ob H .N-ob d = elim - (λ _ → (H ⟅ d ⟆) .snd) - (λ (c , g , a) → (H ⟪ g ⟫) a) - (λ {_ _ (shift g f a) i → H .F-seq (F ⟪ f ⟫) g i a}) + (λ _ → (H ⟅ d ⟆) .snd) + (λ (c , g , a) → (H ⟪ g ⟫) a) + (λ {_ _ (shift g f a) i → H .F-seq (F ⟪ f ⟫) g i a}) ε .N-ob H .N-hom g' = - funExt (elimProp (λ _ → (H ⟅ _ ⟆) .snd _ _) (λ (c , g , a) → funExt⁻ (H .F-seq g g') a)) + funExt (elimProp (λ _ → (H ⟅ _ ⟆) .snd _ _) (λ (c , g , a) → funExt⁻ (H .F-seq g g') a)) ε .N-hom {H} {H'} α = makeNatTransPath (funExt₂ λ d → - elimProp (λ _ → (H' ⟅ _ ⟆) .snd _ _) + elimProp (λ _ → (H' ⟅ _ ⟆) .snd _ _) (λ (c , g , a) → sym (funExt⁻ (α .N-hom g) a))) - Δ₁ : ∀ G → seqTrans (Lan ⟪ η ⟦ G ⟧ ⟫) (ε ⟦ Lan ⟅ G ⟆ ⟧) ≡ idTrans _ + Δ₁ : ∀ G → seqTrans (Lan ⟪ η ⟦ G ⟧ ⟫) (ε ⟦ Lan ⟅ G ⟆ ⟧) ≡ idTrans _ Δ₁ G = makeNatTransPath (funExt₂ λ d → elimProp (λ _ → squash/ _ _) (λ (c , g , a) → [ c , g D.⋆ D.id , a ] - ≡[ i ]⟨ [ c , (g D.⋆ F .F-id (~ i)) , a ] ⟩ - [ c , g D.⋆ (F ⟪ C.id ⟫) , a ] + ≡[ i ]⟨ [ c , (g D.⋆ F .F-id (~ i)) , a ] ⟩ + [ c , g D.⋆ (F ⟪ C.id ⟫) , a ] ≡⟨ shift/ g C.id a ⟩ - [ c , g , (G ⟪ C.id ⟫) a ] - ≡[ i ]⟨ [ c , g , G .F-id i a ] ⟩ + [ c , g , (G ⟪ C.id ⟫) a ] + ≡[ i ]⟨ [ c , g , G .F-id i a ] ⟩ [ c , g , a ] ∎)) - Δ₂ : ∀ H → seqTrans (η ⟦ F* ⟅ H ⟆ ⟧) (F* ⟪ ε ⟦ H ⟧ ⟫) ≡ idTrans _ - Δ₂ H = makeNatTransPath (funExt λ c → H .F-id) + Δ₂ : ∀ H → seqTrans (η ⟦ F* ⟅ H ⟆ ⟧) (F* ⟪ ε ⟦ H ⟧ ⟫) ≡ idTrans _ + Δ₂ H = makeNatTransPath (funExt λ c → H .F-id) - adj : Lan ⊣ F* - adj ._⊣_.η = η - adj ._⊣_.ε = ε - adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₁ = Δ₁ - adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₂ = Δ₂ + adj : Lan ⊣ F* + adj ._⊣_.η = η + adj ._⊣_.ε = ε + adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₁ = Δ₁ + adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₂ = Δ₂ {- Right Kan extension of a functor C → D to a functor PresheafCategory C ℓ → PresheafCategory D ℓ @@ -204,10 +204,10 @@ module Ran {ℓC ℓC' ℓD ℓD'} ℓS {C : Category ℓC ℓC'} {D : Category ℓD ℓD'} - (F : Functor C D) + (F : Functor C D) where - open Functor + open Functor open NatTrans private @@ -221,33 +221,33 @@ -} ℓ = ℓ-max (ℓ-max (ℓ-max ℓC ℓC') ℓD') ℓS - module _ (G : Functor (C ^op) (SET ℓ)) where + module _ (G : Functor (C ^op) (SET ℓ)) where -- Definition of the end record End (d : D.ob) : Type ℓ where field - fun : (c : C.ob) (g : D.Hom[ F ⟅ c ⟆ , d ]) → G .F-ob c .fst - coh : {c c' : C.ob} (f : C.Hom[ c , c' ]) (g : D.Hom[ F ⟅ c' ⟆ , d ]) - → fun c (F ⟪ f ⟫ ⋆⟨ D ⟩ g) ≡ (G ⟪ f ⟫) (fun c' g) + fun : (c : C.ob) (g : D.Hom[ F ⟅ c ⟆ , d ]) → G .F-ob c .fst + coh : {c c' : C.ob} (f : C.Hom[ c , c' ]) (g : D.Hom[ F ⟅ c' ⟆ , d ]) + → fun c (F ⟪ f ⟫ ⋆⟨ D ⟩ g) ≡ (G ⟪ f ⟫) (fun c' g) open End end≡ : {d : D.ob} {x x' : End d} → (∀ c g → x .fun c g ≡ x' .fun c g) → x ≡ x' end≡ h i .fun c g = h c g i end≡ {_} {x} {x'} h i .coh f g = - isSet→isSet' (G .F-ob _ .snd) + isSet→isSet' (G .F-ob _ .snd) (x .coh f g) (x' .coh f g) - (h _ (F ⟪ f ⟫ ⋆⟨ D ⟩ g)) - (cong (G ⟪ f ⟫) (h _ g)) + (h _ (F ⟪ f ⟫ ⋆⟨ D ⟩ g)) + (cong (G ⟪ f ⟫) (h _ g)) i -- Action of End on arrows in D mapR : {d d' : D.ob} (h : D.Hom[ d' , d ]) → End d → End d' mapR h x .fun c g = x .fun c (g ⋆⟨ D ⟩ h) - mapR h x .coh f g = cong (x .fun _) (D.⋆Assoc (F ⟪ f ⟫) g h) ∙ x .coh f (g ⋆⟨ D ⟩ h) + mapR h x .coh f g = cong (x .fun _) (D.⋆Assoc (F ⟪ f ⟫) g h) ∙ x .coh f (g ⋆⟨ D ⟩ h) mapRId : (d : D.ob) → mapR (D.id {x = d}) ≡ idfun (End d) mapRId h = funExt λ x → end≡ λ c g → cong (x .fun c) (D.⋆IdR g) @@ -259,30 +259,30 @@ open End - RanOb : Functor (C ^op) (SET ℓ) → Functor (D ^op) (SET _) - RanOb G .F-ob d .fst = End G d - RanOb G .F-ob d .snd = + RanOb : Functor (C ^op) (SET ℓ) → Functor (D ^op) (SET _) + RanOb G .F-ob d .fst = End G d + RanOb G .F-ob d .snd = -- We use that End is equivalent to a Σ-type to prove its HLevel more easily isOfHLevelRetract 2 {B = - Σ[ z ∈ ((c : C.ob) (g : D.Hom[ F ⟅ c ⟆ , d ]) → G .F-ob c .fst) ] - ({c c' : C.ob} (f : C.Hom[ c , c' ]) (g : D.Hom[ F ⟅ c' ⟆ , d ]) - → z c (F ⟪ f ⟫ ⋆⟨ D ⟩ g) ≡ (G ⟪ f ⟫) (z c' g)) + Σ[ z ∈ ((c : C.ob) (g : D.Hom[ F ⟅ c ⟆ , d ]) → G .F-ob c .fst) ] + ({c c' : C.ob} (f : C.Hom[ c , c' ]) (g : D.Hom[ F ⟅ c' ⟆ , d ]) + → z c (F ⟪ f ⟫ ⋆⟨ D ⟩ g) ≡ (G ⟪ f ⟫) (z c' g)) } (λ x → λ where .fst → x .fun; .snd → x .coh) (λ σ → λ where .fun → σ .fst; .coh → σ .snd) (λ _ → refl) (isSetΣ - (isSetΠ2 λ _ _ → G .F-ob _ .snd) + (isSetΠ2 λ _ _ → G .F-ob _ .snd) (λ _ → isProp→isSet - (isPropImplicitΠ λ _ → isPropImplicitΠ λ _ → isPropΠ2 λ _ _ → G .F-ob _ .snd _ _))) - RanOb G .F-hom = mapR G - RanOb G .F-id {d} = mapRId G d - RanOb G .F-seq h h' = mapR∘ G h' h + (isPropImplicitΠ λ _ → isPropImplicitΠ λ _ → isPropΠ2 λ _ _ → G .F-ob _ .snd _ _))) + RanOb G .F-hom = mapR G + RanOb G .F-id {d} = mapRId G d + RanOb G .F-seq h h' = mapR∘ G h' h -- Action of End on arrows in Cᵒᵖ → Set - module _ {G G' : Functor (C ^op) (SET ℓ)} (α : NatTrans G G') where + module _ {G G' : Functor (C ^op) (SET ℓ)} (α : NatTrans G G') where mapL : (d : D.ob) → End G d → End G' d mapL d x .fun c g = (α ⟦ c ⟧) (x .fun c g) @@ -294,58 +294,58 @@ → mapL d' ∘ mapR G h ≡ mapR G' h ∘ mapL d mapLR h = funExt λ _ → end≡ _ λ _ _ → refl - mapLId : (G : Functor (C ^op) (SET ℓ)) + mapLId : (G : Functor (C ^op) (SET ℓ)) (d : D.ob) → mapL (idTrans G) d ≡ idfun (End G d) mapLId G d = funExt λ _ → end≡ _ λ _ _ → refl - mapL∘ : {G G' G'' : Functor (C ^op) (SET ℓ)} + mapL∘ : {G G' G'' : Functor (C ^op) (SET ℓ)} (β : NatTrans G' G'') (α : NatTrans G G') (d : D.ob) → mapL (seqTrans α β) d ≡ mapL β d ∘ mapL α d mapL∘ β α d = funExt λ _ → end≡ _ λ _ _ → refl - RanHom : {G G' : Functor (C ^op) (SET ℓ)} + RanHom : {G G' : Functor (C ^op) (SET ℓ)} → NatTrans G G' → NatTrans (RanOb G) (RanOb G') RanHom α .N-ob = mapL α RanHom α .N-hom = mapLR α -- Definition of the right Kan extension functor - Ran : Functor (FUNCTOR (C ^op) (SET ℓ)) (FUNCTOR (D ^op) (SET ℓ)) - Ran .F-ob = RanOb - Ran .F-hom = RanHom - Ran .F-id {G} = makeNatTransPath (funExt (mapLId G)) - Ran .F-seq α β = makeNatTransPath (funExt (mapL∘ β α)) + Ran : Functor (FUNCTOR (C ^op) (SET ℓ)) (FUNCTOR (D ^op) (SET ℓ)) + Ran .F-ob = RanOb + Ran .F-hom = RanHom + Ran .F-id {G} = makeNatTransPath (funExt (mapLId G)) + Ran .F-seq α β = makeNatTransPath (funExt (mapL∘ β α)) -- Adjunction between precomposition and right Kan extension private - F* = precomposeF (SET ℓ) (F ^opF) + F* = precomposeF (SET ℓ) (F ^opF) - open UnitCounit + open UnitCounit - η : 𝟙⟨ FUNCTOR (D ^op) (SET ℓ) ⟩ ⇒ (funcComp Ran F*) - η .N-ob G .N-ob d a .fun c g = (G ⟪ g ⟫) a - η .N-ob G .N-ob d a .coh f g = funExt⁻ (G .F-seq g (F ⟪ f ⟫)) a - η .N-ob G .N-hom h = funExt λ a → end≡ _ λ c g → sym (funExt⁻ (G .F-seq h g) a) + η : 𝟙⟨ FUNCTOR (D ^op) (SET ℓ) ⟩ ⇒ (funcComp Ran F*) + η .N-ob G .N-ob d a .fun c g = (G ⟪ g ⟫) a + η .N-ob G .N-ob d a .coh f g = funExt⁻ (G .F-seq g (F ⟪ f ⟫)) a + η .N-ob G .N-hom h = funExt λ a → end≡ _ λ c g → sym (funExt⁻ (G .F-seq h g) a) η .N-hom {G} {G'} α = makeNatTransPath (funExt₂ λ d a → end≡ _ λ c g → sym (funExt⁻ (α .N-hom g) a)) - ε : funcComp F* Ran ⇒ 𝟙⟨ FUNCTOR (C ^op) (SET ℓ) ⟩ + ε : funcComp F* Ran ⇒ 𝟙⟨ FUNCTOR (C ^op) (SET ℓ) ⟩ ε .N-ob H .N-ob c x = x .fun c D.id ε .N-ob H .N-hom {c} {c'} g = funExt λ x → cong (x .fun c') (D.⋆IdL _ ∙ sym (D.⋆IdR _)) ∙ x .coh g D.id ε .N-hom {H} {H'} α = makeNatTransPath refl - Δ₁ : ∀ G → seqTrans (F* ⟪ η ⟦ G ⟧ ⟫) (ε ⟦ F* ⟅ G ⟆ ⟧) ≡ idTrans _ - Δ₁ G = makeNatTransPath (funExt₂ λ c a → funExt⁻ (G .F-id) a) + Δ₁ : ∀ G → seqTrans (F* ⟪ η ⟦ G ⟧ ⟫) (ε ⟦ F* ⟅ G ⟆ ⟧) ≡ idTrans _ + Δ₁ G = makeNatTransPath (funExt₂ λ c a → funExt⁻ (G .F-id) a) - Δ₂ : ∀ H → seqTrans (η ⟦ Ran ⟅ H ⟆ ⟧) (Ran ⟪ ε ⟦ H ⟧ ⟫) ≡ idTrans _ + Δ₂ : ∀ H → seqTrans (η ⟦ Ran ⟅ H ⟆ ⟧) (Ran ⟪ ε ⟦ H ⟧ ⟫) ≡ idTrans _ Δ₂ H = makeNatTransPath (funExt₂ λ c x → end≡ _ λ c' g → cong (x .fun c') (D.⋆IdL g)) - adj : F* ⊣ Ran - adj ._⊣_.η = η - adj ._⊣_.ε = ε - adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₁ = Δ₁ - adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₂ = Δ₂ + adj : F* ⊣ Ran + adj ._⊣_.η = η + adj ._⊣_.ε = ε + adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₁ = Δ₁ + adj ._⊣_.triangleIdentities .TriangleIdentities.Δ₂ = Δ₂ \ No newline at end of file diff --git a/Cubical.Categories.Presheaf.Morphism.html b/Cubical.Categories.Presheaf.Morphism.html index 8083e34e65..c5ba77a033 100644 --- a/Cubical.Categories.Presheaf.Morphism.html +++ b/Cubical.Categories.Presheaf.Morphism.html @@ -39,41 +39,41 @@ open Category open Contravariant -open Functor +open Functor open NatTrans open UniversalElement module _ {C : Category ℓc ℓc'}{D : Category ℓd ℓd'} - (F : Functor C D) + (F : Functor C D) (P : Presheaf C ℓp) (Q : Presheaf D ℓq) where PshHom : Type (ℓ-max (ℓ-max (ℓ-max ℓc ℓc') ℓp) ℓq) PshHom = PresheafCategory C (ℓ-max ℓp ℓq) - [ LiftF {ℓp}{ℓq} ∘F P , LiftF {ℓq}{ℓp} ∘F Q ∘F (F ^opF) ] + [ LiftF {ℓp}{ℓq} ∘F P , LiftF {ℓq}{ℓp} ∘F Q ∘F (F ^opF) ] module _ (h : PshHom) where -- This should define a functor on the category of elements pushElt : Elementᴾ {C = C} P → Elementᴾ {C = D} Q - pushElt (A , η) = (F ⟅ A ⟆) , (h .N-ob A (lift η) .lower) + pushElt (A , η) = (F ⟅ A ⟆) , (h .N-ob A (lift η) .lower) pushEltNat : ∀ {B : C .ob} (η : Elementᴾ {C = C} P) (f : C [ B , η .fst ]) - → (pushElt η .snd ∘ᴾ⟨ D , Q ⟩ F .F-hom f) + → (pushElt η .snd ∘ᴾ⟨ D , Q ⟩ F .F-hom f) ≡ pushElt (B , η .snd ∘ᴾ⟨ C , P ⟩ f) .snd pushEltNat η f i = h .N-hom f (~ i) (lift (η .snd)) .lower - pushEltF : Functor (∫ᴾ_ {C = C} P) (∫ᴾ_ {C = D} Q) - pushEltF .F-ob = pushElt - pushEltF .F-hom {x}{y} (f , commutes) .fst = F .F-hom f - pushEltF .F-hom {x}{y} (f , commutes) .snd = - pushElt _ .snd ∘ᴾ⟨ D , Q ⟩ F .F-hom f + pushEltF : Functor (∫ᴾ_ {C = C} P) (∫ᴾ_ {C = D} Q) + pushEltF .F-ob = pushElt + pushEltF .F-hom {x}{y} (f , commutes) .fst = F .F-hom f + pushEltF .F-hom {x}{y} (f , commutes) .snd = + pushElt _ .snd ∘ᴾ⟨ D , Q ⟩ F .F-hom f ≡⟨ pushEltNat y f ⟩ pushElt (_ , y .snd ∘ᴾ⟨ C , P ⟩ f) .snd ≡⟨ cong (λ a → pushElt a .snd) (ΣPathP (refl , commutes)) ⟩ pushElt x .snd ∎ - pushEltF .F-id = Σ≡Prop (λ x → (Q ⟅ _ ⟆) .snd _ _) (F .F-id) - pushEltF .F-seq f g = - Σ≡Prop ((λ x → (Q ⟅ _ ⟆) .snd _ _)) (F .F-seq (f .fst) (g .fst)) + pushEltF .F-id = Σ≡Prop (λ x → (Q ⟅ _ ⟆) .snd _ _) (F .F-id) + pushEltF .F-seq f g = + Σ≡Prop ((λ x → (Q ⟅ _ ⟆) .snd _ _)) (F .F-seq (f .fst) (g .fst)) preservesRepresentation : ∀ (η : UniversalElement C P) → Type (ℓ-max (ℓ-max ℓd ℓd') ℓq) diff --git a/Cubical.Categories.Presheaf.NonPresheaf.Cofree.html b/Cubical.Categories.Presheaf.NonPresheaf.Cofree.html index 394e218fe7..4b97e50d44 100644 --- a/Cubical.Categories.Presheaf.NonPresheaf.Cofree.html +++ b/Cubical.Categories.Presheaf.NonPresheaf.Cofree.html @@ -13,7 +13,7 @@ open import Cubical.Categories.Presheaf.NonPresheaf.Forget open Category -open Functor +open Functor open NatTrans private @@ -22,19 +22,19 @@ cofreePresheaf : (C : Category ℓ ℓ') → NonPresheaf C ℓS → Presheaf C (ℓ-max (ℓ-max ℓ ℓ') ℓS) -fst (F-ob (cofreePresheaf C X) c) = (b : ob C) → C [ b , c ] → fst (X b) -snd (F-ob (cofreePresheaf C X) c) = isSetΠ (λ b → isSetΠ (λ _ → snd (X b))) -F-hom (cofreePresheaf C X) {d}{c} ψ x b φ = x b (φ ⋆⟨ C ⟩ ψ) -F-id (cofreePresheaf C X) {c} i x b φ = x b (⋆IdR C φ i) -F-seq (cofreePresheaf C X) {e}{d}{c} ω ψ i x b φ = x b (⋆Assoc C φ ψ ω (~ i)) +fst (F-ob (cofreePresheaf C X) c) = (b : ob C) → C [ b , c ] → fst (X b) +snd (F-ob (cofreePresheaf C X) c) = isSetΠ (λ b → isSetΠ (λ _ → snd (X b))) +F-hom (cofreePresheaf C X) {d}{c} ψ x b φ = x b (φ ⋆⟨ C ⟩ ψ) +F-id (cofreePresheaf C X) {c} i x b φ = x b (⋆IdR C φ i) +F-seq (cofreePresheaf C X) {e}{d}{c} ω ψ i x b φ = x b (⋆Assoc C φ ψ ω (~ i)) CofreePresheaf : (C : Category ℓ ℓ') - → Functor (NonPresheafCategory C ℓS) (PresheafCategory C (ℓ-max (ℓ-max ℓ ℓ') ℓS)) -F-ob (CofreePresheaf C) = cofreePresheaf C -N-ob (F-hom (CofreePresheaf C) {X} {Y} f) c x b φ = f b (x b φ) -N-hom (F-hom (CofreePresheaf C) {X} {Y} f) {c}{c'} g = refl -F-id (CofreePresheaf C) {X} = makeNatTransPath refl -F-seq (CofreePresheaf C) {X}{Y}{Z} f g = makeNatTransPath refl + → Functor (NonPresheafCategory C ℓS) (PresheafCategory C (ℓ-max (ℓ-max ℓ ℓ') ℓS)) +F-ob (CofreePresheaf C) = cofreePresheaf C +N-ob (F-hom (CofreePresheaf C) {X} {Y} f) c x b φ = f b (x b φ) +N-hom (F-hom (CofreePresheaf C) {X} {Y} f) {c}{c'} g = refl +F-id (CofreePresheaf C) {X} = makeNatTransPath refl +F-seq (CofreePresheaf C) {X}{Y}{Z} f g = makeNatTransPath refl -- Prove relative adjunction w.r.t. a level-lift between SETs? \ No newline at end of file diff --git a/Cubical.Categories.Presheaf.NonPresheaf.Forget.html b/Cubical.Categories.Presheaf.NonPresheaf.Forget.html index 6ab16cfbba..8384976aac 100644 --- a/Cubical.Categories.Presheaf.NonPresheaf.Forget.html +++ b/Cubical.Categories.Presheaf.NonPresheaf.Forget.html @@ -14,7 +14,7 @@ open import Cubical.Categories.Presheaf.Base open Category -open Functor +open Functor open NatTrans private @@ -28,9 +28,9 @@ → Category (ℓ-max ℓ (ℓ-suc ℓS)) (ℓ-max ℓ ℓS) NonPresheafCategory C ℓS = ΠC (ob C) (λ _ → SET ℓS) -ForgetPresheaf : (C : Category ℓ ℓ') → Functor (PresheafCategory C ℓS) (NonPresheafCategory C ℓS) -F-ob (ForgetPresheaf C) pshX = F-ob pshX -F-hom (ForgetPresheaf C) {pshX} {pshY} pshF = N-ob pshF -F-id (ForgetPresheaf C) {pshX} = refl -F-seq (ForgetPresheaf C) {pshX} {pshY} {pshZ} pshF pshG = refl +ForgetPresheaf : (C : Category ℓ ℓ') → Functor (PresheafCategory C ℓS) (NonPresheafCategory C ℓS) +F-ob (ForgetPresheaf C) pshX = F-ob pshX +F-hom (ForgetPresheaf C) {pshX} {pshY} pshF = N-ob pshF +F-id (ForgetPresheaf C) {pshX} = refl +F-seq (ForgetPresheaf C) {pshX} {pshY} {pshZ} pshF pshG = refl \ No newline at end of file diff --git a/Cubical.Categories.Presheaf.NonPresheaf.Free.html b/Cubical.Categories.Presheaf.NonPresheaf.Free.html index ba67bc8eff..9e3b3936d2 100644 --- a/Cubical.Categories.Presheaf.NonPresheaf.Free.html +++ b/Cubical.Categories.Presheaf.NonPresheaf.Free.html @@ -15,7 +15,7 @@ open import Cubical.Categories.Presheaf.NonPresheaf.Forget open Category -open Functor +open Functor open NatTrans private @@ -24,19 +24,19 @@ freePresheaf : (C : Category ℓ ℓ') → isSet (ob C) → NonPresheaf C ℓS → Presheaf C (ℓ-max (ℓ-max ℓ ℓ') ℓS) -fst (F-ob (freePresheaf C isSetObC X) c) = Σ[ d ∈ ob C ] C [ c , d ] × fst (X d) -snd (F-ob (freePresheaf C isSetObC X) c) = isSetΣ isSetObC λ d → isSet× (isSetHom C) (snd (X d)) -F-hom (freePresheaf C isSetObC X) {d}{c} φ (e , ψ , x) = e , φ ⋆⟨ C ⟩ ψ , x -F-id (freePresheaf C isSetObC X) {c} i (d , φ , x) = d , ⋆IdL C φ i , x -F-seq (freePresheaf C isSetObC X) {e}{d}{c} ψ φ i (f , ω , x) = f , ⋆Assoc C φ ψ ω i , x +fst (F-ob (freePresheaf C isSetObC X) c) = Σ[ d ∈ ob C ] C [ c , d ] × fst (X d) +snd (F-ob (freePresheaf C isSetObC X) c) = isSetΣ isSetObC λ d → isSet× (isSetHom C) (snd (X d)) +F-hom (freePresheaf C isSetObC X) {d}{c} φ (e , ψ , x) = e , φ ⋆⟨ C ⟩ ψ , x +F-id (freePresheaf C isSetObC X) {c} i (d , φ , x) = d , ⋆IdL C φ i , x +F-seq (freePresheaf C isSetObC X) {e}{d}{c} ψ φ i (f , ω , x) = f , ⋆Assoc C φ ψ ω i , x FreePresheaf : (C : Category ℓ ℓ') → isSet (ob C) - → Functor (NonPresheafCategory C ℓS) (PresheafCategory C (ℓ-max (ℓ-max ℓ ℓ') ℓS)) -F-ob (FreePresheaf C isSetObC) X = freePresheaf C isSetObC X -N-ob (F-hom (FreePresheaf C isSetObC) {X} {Y} f) c (d , φ , x) = d , φ , f d x -N-hom (F-hom (FreePresheaf C isSetObC) {X} {Y} f) {c} {c'} g = refl -F-id (FreePresheaf C isSetObC) {X} = makeNatTransPath (funExt (λ c → funExt λ {(d , φ , x) → refl})) -F-seq (FreePresheaf C isSetObC) {X}{Y}{Z} f g = makeNatTransPath (funExt (λ c → funExt λ {(d , φ , x) → refl})) + → Functor (NonPresheafCategory C ℓS) (PresheafCategory C (ℓ-max (ℓ-max ℓ ℓ') ℓS)) +F-ob (FreePresheaf C isSetObC) X = freePresheaf C isSetObC X +N-ob (F-hom (FreePresheaf C isSetObC) {X} {Y} f) c (d , φ , x) = d , φ , f d x +N-hom (F-hom (FreePresheaf C isSetObC) {X} {Y} f) {c} {c'} g = refl +F-id (FreePresheaf C isSetObC) {X} = makeNatTransPath (funExt (λ c → funExt λ {(d , φ , x) → refl})) +F-seq (FreePresheaf C isSetObC) {X}{Y}{Z} f g = makeNatTransPath (funExt (λ c → funExt λ {(d , φ , x) → refl})) -- Prove relative adjunction w.r.t. a level-lift between SETs? \ No newline at end of file diff --git a/Cubical.Categories.Presheaf.Properties.html b/Cubical.Categories.Presheaf.Properties.html index a44d385955..75dc6926d4 100644 --- a/Cubical.Categories.Presheaf.Properties.html +++ b/Cubical.Categories.Presheaf.Properties.html @@ -32,9 +32,9 @@ -- (PresheafCategory C) / F ≃ᶜ PresheafCategory (∫ᴾ F) -module _ {ℓS : Level} (C : Category ℓ ℓ') (F : Functor (C ^op) (SET ℓS)) where +module _ {ℓS : Level} (C : Category ℓ ℓ') (F : Functor (C ^op) (SET ℓS)) where open Category - open Functor + open Functor open _≃ᶜ_ open WeakInverse open NatTrans @@ -49,7 +49,7 @@ → fst a' ≡ fst b' → PathP (λ i → fiber (ϕ ⟦ c ⟧) (px i)) a' b' fibersEqIfRepsEqNatTrans ϕ {c} {x} {x'} {px} {a , fiba} {b , fibb} p - = fibersEqIfRepsEq {isSetB = snd (F ⟅ c ⟆)} (ϕ ⟦ c ⟧) p + = fibersEqIfRepsEq {isSetB = snd (F ⟅ c ⟆)} (ϕ ⟦ c ⟧) p -- ======================================== -- K : Slice → PresheafCategory @@ -58,27 +58,27 @@ -- action on (slice) objects K-ob : (s : SliceCat .ob) → (PresheafCategory (∫ᴾ F) ℓS .ob) -- we take (c , x) to the fiber in A of ϕ over x - K-ob (sliceob {A} ϕ) .F-ob (c , x) + K-ob (sliceob {A} ϕ) .F-ob (c , x) = (fiber (ϕ ⟦ c ⟧) x) - , isOfHLevelΣ 2 (snd (A ⟅ c ⟆)) λ _ → isSet→isGroupoid (snd (F ⟅ c ⟆)) _ _ + , isOfHLevelΣ 2 (snd (A ⟅ c ⟆)) λ _ → isSet→isGroupoid (snd (F ⟅ c ⟆)) _ _ -- for morphisms, we just apply A ⟪ h ⟫ (plus equality proof) - K-ob (sliceob {A} ϕ) .F-hom {d , y} {c , x} (h , com) (b , eq) - = ((A ⟪ h ⟫) b) - , ((ϕ ⟦ c ⟧) ((A ⟪ h ⟫) b) + K-ob (sliceob {A} ϕ) .F-hom {d , y} {c , x} (h , com) (b , eq) + = ((A ⟪ h ⟫) b) + , ((ϕ ⟦ c ⟧) ((A ⟪ h ⟫) b) ≡[ i ]⟨ (ϕ .N-hom h) i b ⟩ - (F ⟪ h ⟫) ((ϕ ⟦ d ⟧) b) - ≡[ i ]⟨ (F ⟪ h ⟫) (eq i) ⟩ - (F ⟪ h ⟫) y + (F ⟪ h ⟫) ((ϕ ⟦ d ⟧) b) + ≡[ i ]⟨ (F ⟪ h ⟫) (eq i) ⟩ + (F ⟪ h ⟫) y ≡⟨ com ⟩ x ∎) -- functoriality follows from functoriality of A - K-ob (sliceob {A} ϕ) .F-id {x = (c , x)} + K-ob (sliceob {A} ϕ) .F-id {x = (c , x)} = funExt λ { (a , fibp) - → fibersEqIfRepsEqNatTrans ϕ (λ i → A .F-id i a) } - K-ob (sliceob {A} ϕ) .F-seq {x = (c , x)} {(d , y)} {(e , z)} (f' , eq1) (g' , eq2) + → fibersEqIfRepsEqNatTrans ϕ (λ i → A .F-id i a) } + K-ob (sliceob {A} ϕ) .F-seq {x = (c , x)} {(d , y)} {(e , z)} (f' , eq1) (g' , eq2) = funExt λ { ( a , fibp ) - → fibersEqIfRepsEqNatTrans ϕ (λ i → (A .F-seq f' g') i a) } + → fibersEqIfRepsEqNatTrans ϕ (λ i → (A .F-seq f' g') i a) } -- action on morphisms (in this case, natural transformation) @@ -92,29 +92,29 @@ -- just apply the natural transformation (ε) we're given -- this ensures that we stay in the fiber over x due to the commutativity given by slicenesss - η-ob : (el : (∫ᴾ F) .ob) → (fst (P ⟅ el ⟆) → fst (Q ⟅ el ⟆) ) + η-ob : (el : (∫ᴾ F) .ob) → (fst (P ⟅ el ⟆) → fst (Q ⟅ el ⟆) ) η-ob (c , x) (a , ϕa≡x) = ((ε ⟦ c ⟧) a) , εψ≡ϕ ∙ ϕa≡x where εψ≡ϕ : (ψ ⟦ c ⟧) ((ε ⟦ c ⟧) a) ≡ (ϕ ⟦ c ⟧) a εψ≡ϕ i = ((com i) ⟦ c ⟧) a - η-hom : ∀ {el1 el2} (h : (∫ᴾ F) [ el1 , el2 ]) (ae : fst (P ⟅ el2 ⟆)) → η-ob el1 ((P ⟪ h ⟫) ae) ≡ (Q ⟪ h ⟫) (η-ob el2 ae) + η-hom : ∀ {el1 el2} (h : (∫ᴾ F) [ el1 , el2 ]) (ae : fst (P ⟅ el2 ⟆)) → η-ob el1 ((P ⟪ h ⟫) ae) ≡ (Q ⟪ h ⟫) (η-ob el2 ae) η-hom {el1 = (c , x)} {d , y} (h , eqh) (a , eqa) = fibersEqIfRepsEqNatTrans ψ (λ i → ε .N-hom h i a) - K : Functor SliceCat (PresheafCategory (∫ᴾ F) ℓS) - K .F-ob = K-ob - K .F-hom = K-hom - K .F-id = makeNatTransPath + K : Functor SliceCat (PresheafCategory (∫ᴾ F) ℓS) + K .F-ob = K-ob + K .F-hom = K-hom + K .F-id = makeNatTransPath (funExt λ cx@(c , x) → funExt λ aeq@(a , eq) - → fibersEqIfRepsEq {isSetB = snd (F ⟅ c ⟆)} _ refl) - K .F-seq (slicehom α eqa) (slicehom β eqb) + → fibersEqIfRepsEq {isSetB = snd (F ⟅ c ⟆)} _ refl) + K .F-seq (slicehom α eqa) (slicehom β eqb) = makeNatTransPath (funExt λ cx@(c , x) → funExt λ aeq@(a , eq) - → fibersEqIfRepsEq {isSetB = snd (F ⟅ c ⟆)} _ refl) + → fibersEqIfRepsEq {isSetB = snd (F ⟅ c ⟆)} _ refl) -- ======================================== @@ -128,44 +128,44 @@ where -- sends c to the disjoint union of all the images under P LF-ob : (c : C .ob) → (SET _) .ob - LF-ob c = (Σ[ x ∈ fst (F ⟅ c ⟆) ] fst (P ⟅ c , x ⟆)) , isSetΣ (snd (F ⟅ c ⟆)) (λ x → snd (P ⟅ c , x ⟆)) + LF-ob c = (Σ[ x ∈ fst (F ⟅ c ⟆) ] fst (P ⟅ c , x ⟆)) , isSetΣ (snd (F ⟅ c ⟆)) (λ x → snd (P ⟅ c , x ⟆)) -- defines a function piecewise over the fibers by applying P LF-hom : ∀ {x y} → (f : C [ y , x ]) → (SET _) [ LF-ob x , LF-ob y ] - LF-hom {x = c} {d} f (x , a) = ((F ⟪ f ⟫) x) , (P ⟪ f , refl ⟫) a + LF-hom {x = c} {d} f (x , a) = ((F ⟪ f ⟫) x) , (P ⟪ f , refl ⟫) a - L-ob-ob : Functor (C ^op) (SET _) - L-ob-ob .F-ob = LF-ob - L-ob-ob .F-hom = LF-hom - L-ob-ob .F-id {x = c} + L-ob-ob : Functor (C ^op) (SET _) + L-ob-ob .F-ob = LF-ob + L-ob-ob .F-hom = LF-hom + L-ob-ob .F-id {x = c} = funExt idFunExt where idFunExt : ∀ (un : fst (LF-ob c)) → (LF-hom (C .id) un) ≡ un idFunExt (x , X) = ΣPathP (leftEq , rightEq) where - leftEq : (F ⟪ C .id ⟫) x ≡ x - leftEq i = F .F-id i x + leftEq : (F ⟪ C .id ⟫) x ≡ x + leftEq i = F .F-id i x - rightEq : PathP (λ i → fst (P ⟅ c , leftEq i ⟆)) - ((P ⟪ C .id , refl ⟫) X) X + rightEq : PathP (λ i → fst (P ⟅ c , leftEq i ⟆)) + ((P ⟪ C .id , refl ⟫) X) X rightEq = left ▷ right where -- the id morphism in (∫ᴾ F) - ∫id = C .id , funExt⁻ (F .F-id) x + ∫id = C .id , funExt⁻ (F .F-id) x -- functoriality of P gives us close to what we want - right : (P ⟪ ∫id ⟫) X ≡ X - right i = P .F-id i X + right : (P ⟪ ∫id ⟫) X ≡ X + right i = P .F-id i X -- but need to do more work to show that (C .id , refl) ≡ ∫id - left : PathP (λ i → fst (P ⟅ c , leftEq i ⟆)) - ((P ⟪ C .id , refl ⟫) X) - ((P ⟪ ∫id ⟫) X) - left i = (P ⟪ ∫ᴾhomEq {F = F} (C .id , refl) ∫id (λ i → (c , leftEq i)) refl refl i ⟫) X - L-ob-ob .F-seq {x = c} {d} {e} f g + left : PathP (λ i → fst (P ⟅ c , leftEq i ⟆)) + ((P ⟪ C .id , refl ⟫) X) + ((P ⟪ ∫id ⟫) X) + left i = (P ⟪ ∫ᴾhomEq {F = F} (C .id , refl) ∫id (λ i → (c , leftEq i)) refl refl i ⟫) X + L-ob-ob .F-seq {x = c} {d} {e} f g = funExt seqFunEq where seqFunEq : ∀ (un : fst (LF-ob c)) @@ -175,26 +175,26 @@ -- the left component is comparing the action of F on x -- equality follows from functoriality of F -- leftEq : fst (LF-hom (g ⋆⟨ C ⟩ f) un) ≡ fst ((LF-hom g) (LF-hom f un)) - leftEq : (F ⟪ g ⋆⟨ C ⟩ f ⟫) x ≡ (F ⟪ g ⟫) ((F ⟪ f ⟫) x) - leftEq i = F .F-seq f g i x + leftEq : (F ⟪ g ⋆⟨ C ⟩ f ⟫) x ≡ (F ⟪ g ⟫) ((F ⟪ f ⟫) x) + leftEq i = F .F-seq f g i x -- on the right, equality also follows from functoriality of P -- but it's more complicated because of heterogeneity -- since leftEq is not a definitional equality - rightEq : PathP (λ i → fst (P ⟅ e , leftEq i ⟆)) - ((P ⟪ g ⋆⟨ C ⟩ f , refl ⟫) X) - ((P ⟪ g , refl ⟫) ((P ⟪ f , refl ⟫) X)) + rightEq : PathP (λ i → fst (P ⟅ e , leftEq i ⟆)) + ((P ⟪ g ⋆⟨ C ⟩ f , refl ⟫) X) + ((P ⟪ g , refl ⟫) ((P ⟪ f , refl ⟫) X)) rightEq = left ▷ right where -- functoriality of P only gets us to this weird composition on the left - right : (P ⟪ (g , refl) ⋆⟨ (∫ᴾ F) ⟩ (f , refl) ⟫) X ≡ (P ⟪ g , refl ⟫) ((P ⟪ f , refl ⟫) X) - right i = P .F-seq (f , refl) (g , refl) i X + right : (P ⟪ (g , refl) ⋆⟨ (∫ᴾ F) ⟩ (f , refl) ⟫) X ≡ (P ⟪ g , refl ⟫) ((P ⟪ f , refl ⟫) X) + right i = P .F-seq (f , refl) (g , refl) i X -- so we need to show that this composition is actually equal to the one we want - left : PathP (λ i → fst (P ⟅ e , leftEq i ⟆)) - ((P ⟪ g ⋆⟨ C ⟩ f , refl ⟫) X) - ((P ⟪ (g , refl) ⋆⟨ (∫ᴾ F) ⟩ (f , refl) ⟫) X) - left i = (P ⟪ ∫ᴾhomEq {F = F} (g ⋆⟨ C ⟩ f , refl) ((g , refl) ⋆⟨ (∫ᴾ F) ⟩ (f , refl)) (λ i → (e , leftEq i)) refl refl i ⟫) X + left : PathP (λ i → fst (P ⟅ e , leftEq i ⟆)) + ((P ⟪ g ⋆⟨ C ⟩ f , refl ⟫) X) + ((P ⟪ (g , refl) ⋆⟨ (∫ᴾ F) ⟩ (f , refl) ⟫) X) + left i = (P ⟪ ∫ᴾhomEq {F = F} (g ⋆⟨ C ⟩ f , refl) ((g , refl) ⋆⟨ (∫ᴾ F) ⟩ (f , refl)) (λ i → (e , leftEq i)) refl refl i ⟫) X L-ob-hom : L-ob-ob ⇒ F L-ob-hom .N-ob c (x , _) = x L-ob-hom .N-hom f = funExt λ (x , _) → refl @@ -213,9 +213,9 @@ arr .N-ob c (x , X) = x , ((η ⟦ c , x ⟧) X) arr .N-hom {c} {d} f = funExt natu where - natuType : fst (A ⟅ c ⟆) → Type _ - natuType xX@(x , X) = ((F ⟪ f ⟫) x , (η ⟦ d , (F ⟪ f ⟫) x ⟧) ((P ⟪ f , refl ⟫) X)) ≡ ((F ⟪ f ⟫) x , (Q ⟪ f , refl ⟫) ((η ⟦ c , x ⟧) X)) - natu : ∀ (xX : fst (A ⟅ c ⟆)) → natuType xX + natuType : fst (A ⟅ c ⟆) → Type _ + natuType xX@(x , X) = ((F ⟪ f ⟫) x , (η ⟦ d , (F ⟪ f ⟫) x ⟧) ((P ⟪ f , refl ⟫) X)) ≡ ((F ⟪ f ⟫) x , (Q ⟪ f , refl ⟫) ((η ⟦ c , x ⟧) X)) + natu : ∀ (xX : fst (A ⟅ c ⟆)) → natuType xX natu (x , X) = ΣPathP (refl , λ i → (η .N-hom (f , refl) i) X) com : arr ⋆⟨ PresheafCategory C ℓS ⟩ ψ ≡ ϕ @@ -225,11 +225,11 @@ → (arr ●ᵛ ψ) ⟦ c ⟧ ≡ ϕ ⟦ c ⟧ comFunExt c = funExt λ x → refl - L : Functor (PresheafCategory (∫ᴾ F) ℓS) SliceCat - L .F-ob = L-ob - L .F-hom = L-hom - L .F-id {cx} = SliceHom-≡-intro' (makeNatTransPath (funExt λ c → refl)) - L .F-seq {cx} {dy} P Q = SliceHom-≡-intro' (makeNatTransPath (funExt λ c → refl)) + L : Functor (PresheafCategory (∫ᴾ F) ℓS) SliceCat + L .F-ob = L-ob + L .F-hom = L-hom + L .F-id {cx} = SliceHom-≡-intro' (makeNatTransPath (funExt λ c → refl)) + L .F-seq {cx} {dy} P Q = SliceHom-≡-intro' (makeNatTransPath (funExt λ c → refl)) -- ======================================== -- η : 𝟙 ≅ LK @@ -251,18 +251,18 @@ -- the natural transformation -- just applies typeSectionIso - ηTrans : 𝟙⟨ SliceCat ⟩ ⇒ (L ∘F K) + ηTrans : 𝟙⟨ SliceCat ⟩ ⇒ (L ∘F K) ηTrans .N-ob sob@(sliceob {A} ϕ) = slicehom A⇒LK comm where - LKA = S-ob (L ⟅ K ⟅ sob ⟆ ⟆) - ψ = S-arr (L ⟅ K ⟅ sob ⟆ ⟆) + LKA = S-ob (L ⟅ K ⟅ sob ⟆ ⟆) + ψ = S-arr (L ⟅ K ⟅ sob ⟆ ⟆) A⇒LK : A ⇒ LKA - A⇒LK .N-ob c = typeSectionIso {isSetB = snd (F ⟅ c ⟆)} (ϕ ⟦ c ⟧) .fun + A⇒LK .N-ob c = typeSectionIso {isSetB = snd (F ⟅ c ⟆)} (ϕ ⟦ c ⟧) .fun A⇒LK .N-hom {c} {d} f = funExt homFunExt where - homFunExt : (x : fst (A ⟅ c ⟆)) - → (((ϕ ⟦ d ⟧) ((A ⟪ f ⟫) x)) , ((A ⟪ f ⟫) x , refl)) ≡ ((F ⟪ f ⟫) ((ϕ ⟦ c ⟧) x) , (A ⟪ f ⟫) x , _) + homFunExt : (x : fst (A ⟅ c ⟆)) + → (((ϕ ⟦ d ⟧) ((A ⟪ f ⟫) x)) , ((A ⟪ f ⟫) x , refl)) ≡ ((F ⟪ f ⟫) ((ϕ ⟦ c ⟧) x) , (A ⟪ f ⟫) x , _) homFunExt x = ΣPathP ((λ i → (ϕ .N-hom f i) x) , fibersEqIfRepsEqNatTrans ϕ refl) comm : (A⇒LK) ●ᵛ ψ ≡ ϕ @@ -270,7 +270,7 @@ ηTrans .N-hom {sliceob {A} α} {sliceob {B} β} (slicehom ϕ eq) = SliceHom-≡-intro' (makeNatTransPath (funExt (λ c → funExt λ a → natFunExt c a))) where - natFunExt : ∀ (c : C .ob) (a : fst (A ⟅ c ⟆)) + natFunExt : ∀ (c : C .ob) (a : fst (A ⟅ c ⟆)) → ((β ⟦ c ⟧) ((ϕ ⟦ c ⟧) a) , (ϕ ⟦ c ⟧) a , _) ≡ ((α ⟦ c ⟧) a , (ϕ ⟦ c ⟧) a , _) natFunExt c a = ΣPathP ((λ i → ((eq i) ⟦ c ⟧) a) , fibersEqIfRepsEqNatTrans β refl) @@ -281,7 +281,7 @@ where isIsoCf : ∀ (c : C .ob) → isIsoC _ (ηTrans .N-ob sob .S-hom ⟦ c ⟧) - isIsoCf c = Morphism.CatIso→isIso (Iso→CatIso (typeSectionIso {isSetB = snd (F ⟅ c ⟆)} (ϕ ⟦ c ⟧))) + isIsoCf c = Morphism.CatIso→isIso (Iso→CatIso (typeSectionIso {isSetB = snd (F ⟅ c ⟆)} (ϕ ⟦ c ⟧))) -- ======================================== @@ -302,66 +302,66 @@ -- the natural isomorphism -- applies typeFiberIso (inv) - εTrans : (K ∘F L) ⇒ 𝟙⟨ PresheafCategory (∫ᴾ F) ℓS ⟩ + εTrans : (K ∘F L) ⇒ 𝟙⟨ PresheafCategory (∫ᴾ F) ℓS ⟩ εTrans .N-ob P = natTrans γ-ob (λ f → funExt (λ a → γ-homFunExt f a)) where - KLP = K ⟅ L ⟅ P ⟆ ⟆ + KLP = K ⟅ L ⟅ P ⟆ ⟆ γ-ob : (el : (∫ᴾ F) .ob) - → (fst (KLP ⟅ el ⟆) → fst (P ⟅ el ⟆) ) - γ-ob el@(c , _) = typeFiberIso {isSetA = snd (F ⟅ c ⟆)} (λ x → fst (P ⟅ c , x ⟆)) .inv + → (fst (KLP ⟅ el ⟆) → fst (P ⟅ el ⟆) ) + γ-ob el@(c , _) = typeFiberIso {isSetA = snd (F ⟅ c ⟆)} (λ x → fst (P ⟅ c , x ⟆)) .inv -- naturality -- the annoying part is all the substs γ-homFunExt : ∀ {el2 el1} → (f' : (∫ᴾ F) [ el2 , el1 ]) - → (∀ (a : fst (KLP ⟅ el1 ⟆)) → γ-ob el2 ((KLP ⟪ f' ⟫) a) ≡ (P ⟪ f' ⟫) (γ-ob el1 a)) + → (∀ (a : fst (KLP ⟅ el1 ⟆)) → γ-ob el2 ((KLP ⟪ f' ⟫) a) ≡ (P ⟪ f' ⟫) (γ-ob el1 a)) γ-homFunExt {d , y} {c , x} f'@(f , comm) a@((x' , X') , eq) i - = comp (λ j → fst (P ⟅ d , eq' j ⟆)) (λ j → λ { (i = i0) → left j - ; (i = i1) → right j }) ((P ⟪ f , refl ⟫) X') + = comp (λ j → fst (P ⟅ d , eq' j ⟆)) (λ j → λ { (i = i0) → left j + ; (i = i1) → right j }) ((P ⟪ f , refl ⟫) X') where -- fiber equality proof that we get from an application of KLP - eq' = snd ((KLP ⟪ f' ⟫) a) + eq' = snd ((KLP ⟪ f' ⟫) a) -- top right of the commuting diagram -- "remove" the subst from the inside - right : PathP (λ i → fst (P ⟅ d , eq' i ⟆)) ((P ⟪ f , refl ⟫) X') ((P ⟪ f , comm ⟫) (subst _ eq X')) - right i = (P ⟪ f , refl≡comm i ⟫) (X'≡subst i) + right : PathP (λ i → fst (P ⟅ d , eq' i ⟆)) ((P ⟪ f , refl ⟫) X') ((P ⟪ f , comm ⟫) (subst _ eq X')) + right i = (P ⟪ f , refl≡comm i ⟫) (X'≡subst i) where - refl≡comm : PathP (λ i → (F ⟪ f ⟫) (eq i) ≡ (eq' i)) refl comm - refl≡comm = isOfHLevel→isOfHLevelDep 1 (λ (v , w) → snd (F ⟅ d ⟆) ((F ⟪ f ⟫) w) v) refl comm λ i → (eq' i , eq i) + refl≡comm : PathP (λ i → (F ⟪ f ⟫) (eq i) ≡ (eq' i)) refl comm + refl≡comm = isOfHLevel→isOfHLevelDep 1 (λ (v , w) → snd (F ⟅ d ⟆) ((F ⟪ f ⟫) w) v) refl comm λ i → (eq' i , eq i) - X'≡subst : PathP (λ i → fst (P ⟅ c , eq i ⟆)) X' (subst _ eq X') - X'≡subst = transport-filler (λ i → fst (P ⟅ c , eq i ⟆)) X' + X'≡subst : PathP (λ i → fst (P ⟅ c , eq i ⟆)) X' (subst _ eq X') + X'≡subst = transport-filler (λ i → fst (P ⟅ c , eq i ⟆)) X' -- bottom left of the commuting diagram -- "remove" the subst from the outside - left : PathP (λ i → fst (P ⟅ d , eq' i ⟆)) ((P ⟪ f , refl ⟫) X') (subst (λ v → fst (P ⟅ d , v ⟆)) eq' ((P ⟪ f , refl ⟫) X')) - left = transport-filler (λ i → fst (P ⟅ d , eq' i ⟆)) ((P ⟪ f , refl ⟫) X') + left : PathP (λ i → fst (P ⟅ d , eq' i ⟆)) ((P ⟪ f , refl ⟫) X') (subst (λ v → fst (P ⟅ d , v ⟆)) eq' ((P ⟪ f , refl ⟫) X')) + left = transport-filler (λ i → fst (P ⟅ d , eq' i ⟆)) ((P ⟪ f , refl ⟫) X') εTrans .N-hom {P} {Q} α = makeNatTransPath (funExt λ cx → funExt λ xX' → ε-homFunExt cx xX') where - KLP = K ⟅ L ⟅ P ⟆ ⟆ + KLP = K ⟅ L ⟅ P ⟆ ⟆ -- naturality of the above construction applies a similar argument as in `γ-homFunExt` - ε-homFunExt : ∀ (cx@(c , x) : (∫ᴾ F) .ob) (xX'@((x' , X') , eq) : fst (KLP ⟅ cx ⟆)) - → subst (λ v → fst (Q ⟅ c , v ⟆)) (snd ((K ⟪ L ⟪ α ⟫ ⟫ ⟦ cx ⟧) xX')) ((α ⟦ c , x' ⟧) X') + ε-homFunExt : ∀ (cx@(c , x) : (∫ᴾ F) .ob) (xX'@((x' , X') , eq) : fst (KLP ⟅ cx ⟆)) + → subst (λ v → fst (Q ⟅ c , v ⟆)) (snd ((K ⟪ L ⟪ α ⟫ ⟫ ⟦ cx ⟧) xX')) ((α ⟦ c , x' ⟧) X') ≡ (α ⟦ c , x ⟧) (subst _ eq X') ε-homFunExt cx@(c , x) xX'@((x' , X') , eq) i - = comp (λ j → fst (Q ⟅ c , eq j ⟆)) (λ j → λ { (i = i0) → left j + = comp (λ j → fst (Q ⟅ c , eq j ⟆)) (λ j → λ { (i = i0) → left j ; (i = i1) → right j }) ((α ⟦ c , x' ⟧) X') where eq' : x' ≡ x - eq' = snd ((K ⟪ L ⟪ α ⟫ ⟫ ⟦ cx ⟧) xX') + eq' = snd ((K ⟪ L ⟪ α ⟫ ⟫ ⟦ cx ⟧) xX') - right : PathP (λ i → fst (Q ⟅ c , eq i ⟆)) ((α ⟦ c , x' ⟧) X') ((α ⟦ c , x ⟧) (subst _ eq X')) + right : PathP (λ i → fst (Q ⟅ c , eq i ⟆)) ((α ⟦ c , x' ⟧) X') ((α ⟦ c , x ⟧) (subst _ eq X')) right i = (α ⟦ c , eq i ⟧) (X'≡subst i) where -- this is exactly the same as the one from before, can refactor? - X'≡subst : PathP (λ i → fst (P ⟅ c , eq i ⟆)) X' (subst _ eq X') + X'≡subst : PathP (λ i → fst (P ⟅ c , eq i ⟆)) X' (subst _ eq X') X'≡subst = transport-filler _ _ -- extracted out type since need to use in in 'left' body as well leftTy : (x' ≡ x) → Type _ - leftTy eq* = PathP (λ i → fst (Q ⟅ c , eq* i ⟆)) ((α ⟦ c , x' ⟧) X') (subst (λ v → fst (Q ⟅ c , v ⟆)) eq' ((α ⟦ c , x' ⟧) X')) + leftTy eq* = PathP (λ i → fst (Q ⟅ c , eq* i ⟆)) ((α ⟦ c , x' ⟧) X') (subst (λ v → fst (Q ⟅ c , v ⟆)) eq' ((α ⟦ c , x' ⟧) X')) left : leftTy eq left = subst @@ -370,7 +370,7 @@ (transport-filler _ _) where eq'≡eq : eq' ≡ eq - eq'≡eq = snd (F ⟅ c ⟆) _ _ eq' eq + eq'≡eq = snd (F ⟅ c ⟆) _ _ eq' eq εIso : ∀ (P : PresheafCategory (∫ᴾ F) ℓS .ob) → isIsoC (PresheafCategory (∫ᴾ F) ℓS) (εTrans ⟦ P ⟧) @@ -378,7 +378,7 @@ where isIsoC' : ∀ (cx : (∫ᴾ F) .ob) → isIsoC (SET _) ((εTrans ⟦ P ⟧) ⟦ cx ⟧) - isIsoC' cx@(c , _) = Morphism.CatIso→isIso (Iso→CatIso (invIso (typeFiberIso {isSetA = snd (F ⟅ c ⟆)} _))) + isIsoC' cx@(c , _) = Morphism.CatIso→isIso (Iso→CatIso (invIso (typeFiberIso {isSetA = snd (F ⟅ c ⟆)} _))) -- putting it all together @@ -397,5 +397,5 @@ -- Isomorphism between presheaves of different levels PshIso : (C : Category ℓ ℓ') (P : Presheaf C ℓS) (Q : Presheaf C ℓS') → Type _ PshIso {ℓS = ℓS}{ℓS' = ℓS'} C P Q = - NatIso (LiftF {ℓ = ℓS}{ℓ' = ℓS'} ∘F P) (LiftF {ℓ = ℓS'}{ℓ' = ℓS} ∘F Q) + NatIso (LiftF {ℓ = ℓS}{ℓ' = ℓS'} ∘F P) (LiftF {ℓ = ℓS'}{ℓ' = ℓS} ∘F Q) \ No newline at end of file diff --git a/Cubical.Categories.Presheaf.Representable.html b/Cubical.Categories.Presheaf.Representable.html index 7f4405e473..39099d6732 100644 --- a/Cubical.Categories.Presheaf.Representable.html +++ b/Cubical.Categories.Presheaf.Representable.html @@ -72,7 +72,7 @@ hasTerminalElement : Type (ℓ-max (ℓ-max ℓo ℓh) ℓp) hasTerminalElement = ∥ TerminalElement ∥₁ - isUniversal : (vertex : C .ob) (element : (P ⟅ vertex ⟆) .fst) + isUniversal : (vertex : C .ob) (element : (P ⟅ vertex ⟆) .fst) → Type (ℓ-max (ℓ-max ℓo ℓh) ℓp) isUniversal vertex element = ∀ A → isEquiv λ (f : C [ A , vertex ]) → element ∘ᴾ⟨ C , P ⟩ f @@ -83,7 +83,7 @@ record UniversalElement : Type (ℓ-max (ℓ-max ℓo ℓh) ℓp) where field vertex : C .ob - element : (P ⟅ vertex ⟆) .fst + element : (P ⟅ vertex ⟆) .fst universal : isUniversal vertex element unquoteDecl UniversalElementIsoΣ = @@ -115,7 +115,7 @@ .Iso.leftInv (repr .snd .trans) (~ i) ⟦ A ⟧) (lift f) .lower) - anIso : Iso (C [ A , repr .fst ]) (fst (Functor.F-ob P A)) + anIso : Iso (C [ A , repr .fst ]) (fst (Functor.F-ob P A)) anIso .Iso.fun f = (repr .snd .trans ⟦ A ⟧) (lift f) .lower anIso .Iso.inv p = repr .snd .nIso A .inv (lift p) .lower anIso .Iso.rightInv b = @@ -139,7 +139,7 @@ Representation≅UniversalElement .Iso.inv = universalElementToRepresentation Representation≅UniversalElement .Iso.rightInv η = isoFunInjective UniversalElementIsoΣ _ _ - (ΣPathP (refl , (Σ≡Prop (λ _ → isPropIsUniversal _ _) + (ΣPathP (refl , (Σ≡Prop (λ _ → isPropIsUniversal _ _) (yonedaᴾ* {C = C} P (η .vertex) .Iso.rightInv (η .element))))) Representation≅UniversalElement .Iso.leftInv repr = ΣPathP (refl , @@ -153,11 +153,11 @@ isTerminalToIsUniversal {η} term A .equiv-proof ϕ .fst .snd = term (_ , ϕ) .fst .snd isTerminalToIsUniversal {η} term A .equiv-proof ϕ .snd (f , commutes) = - Σ≡Prop (λ _ → (P ⟅ A ⟆) .snd _ _) + Σ≡Prop (λ _ → (P ⟅ A ⟆) .snd _ _) (cong fst (term (A , ϕ) .snd (f , commutes))) isUniversalToIsTerminal : - ∀ (vertex : C .ob) (element : (P ⟅ vertex ⟆) .fst) + ∀ (vertex : C .ob) (element : (P ⟅ vertex ⟆) .fst) → isUniversal vertex element → isTerminal Elements (vertex , element) isUniversalToIsTerminal vertex element universal ϕ .fst .fst = @@ -165,8 +165,8 @@ isUniversalToIsTerminal vertex element universal ϕ .fst .snd = universal _ .equiv-proof (ϕ .snd) .fst .snd isUniversalToIsTerminal vertex element universal ϕ .snd (f , commutes) = - Σ≡Prop - (λ _ → (P ⟅ _ ⟆) .snd _ _) + Σ≡Prop + (λ _ → (P ⟅ _ ⟆) .snd _ _) (cong fst (universal _ .equiv-proof (ϕ .snd) .snd (f , commutes))) terminalElementToUniversalElement : TerminalElement → UniversalElement @@ -186,9 +186,9 @@ TerminalElement≅UniversalElement .Iso.inv = universalElementToTerminalElement TerminalElement≅UniversalElement .Iso.rightInv η = isoFunInjective UniversalElementIsoΣ _ _ - (ΣPathP (refl , (Σ≡Prop (λ _ → isPropIsUniversal _ _) refl))) + (ΣPathP (refl , (Σ≡Prop (λ _ → isPropIsUniversal _ _) refl))) TerminalElement≅UniversalElement .Iso.leftInv η = - Σ≡Prop (isPropIsTerminal Elements) refl + Σ≡Prop (isPropIsTerminal Elements) refl Representation≅TerminalElement : Iso Representation TerminalElement Representation≅TerminalElement = diff --git a/Cubical.Categories.Profunctor.Base.html b/Cubical.Categories.Profunctor.Base.html index 410fd671d0..0fd33a948b 100644 --- a/Cubical.Categories.Profunctor.Base.html +++ b/Cubical.Categories.Profunctor.Base.html @@ -83,23 +83,23 @@ _⋆D_ = D._⋆_ Het[_,_] : C.ob → D.ob → Type ℓ' - Het[ c , d ] = ⟨ asFunc ⟅ c , d ⟆ ⟩ + Het[ c , d ] = ⟨ asFunc ⟅ c , d ⟆ ⟩ _⋆L⟨_⟩⋆R_ : ∀ {c c' d' d} → (f : C.Hom[ c , c' ])(r : Het[ c' , d' ])(g : D.Hom[ d' , d ]) → Het[ c , d ] - f ⋆L⟨ r ⟩⋆R g = Functor.F-hom R (f , g) r + f ⋆L⟨ r ⟩⋆R g = Functor.F-hom R (f , g) r ⋆L⟨⟩⋆RAssoc : ∀ {c c' c'' d'' d' d} → (f : C.Hom[ c , c' ]) (f' : C.Hom[ c' , c'' ]) (r : Het[ c'' , d'' ]) (g' : D.Hom[ d'' , d' ]) (g : D.Hom[ d' , d ]) → (f ⋆C f') ⋆L⟨ r ⟩⋆R (g' ⋆D g) ≡ f ⋆L⟨ f' ⋆L⟨ r ⟩⋆R g' ⟩⋆R g - ⋆L⟨⟩⋆RAssoc f f' r g' g i = Functor.F-seq R (f' , g') (f , g) i r + ⋆L⟨⟩⋆RAssoc f f' r g' g i = Functor.F-seq R (f' , g') (f , g) i r ⋆L⟨⟩⋆RId : ∀ {c d} → (r : Het[ c , d ]) → C.id ⋆L⟨ r ⟩⋆R D.id ≡ r - ⋆L⟨⟩⋆RId r i = Functor.F-id R i r + ⋆L⟨⟩⋆RId r i = Functor.F-id R i r _⋆L_ : {c c' : C.ob} {d : D.ob} @@ -191,29 +191,29 @@ swap : {C D : Cat} → C ⊶ D → (D ^op) ⊶ (C ^op) swap R = fromFunc - record { F-ob = λ (d , c) → R.F-ob (c , d) - ; F-hom = λ (f , g) → R.F-hom (g , f) - ; F-id = R.F-id - ; F-seq = λ (fl , fr) (gl , gr) → R.F-seq (fr , fl) (gr , gl) + record { F-ob = λ (d , c) → R.F-ob (c , d) + ; F-hom = λ (f , g) → R.F-hom (g , f) + ; F-id = R.F-id + ; F-seq = λ (fl , fr) (gl , gr) → R.F-seq (fr , fl) (gr , gl) } - where module R = Functor (Profunctor⊶.asFunc R) + where module R = Functor (Profunctor⊶.asFunc R) HomProf : (C : Cat) → C ⊶ C HomProf C = fromFunc (HomFunctor C) _profF⊶[_,_] : {B C D E : Cat} → (R : D ⊶ E) - → (F : Functor B D) - → (G : Functor C E) + → (F : Functor B D) + → (G : Functor C E) → B ⊶ C - R profF⊶[ F , G ] = fromFunc ((Profunctor⊶.asFunc R) ∘F ((F ^opF) ×F G)) + R profF⊶[ F , G ] = fromFunc ((Profunctor⊶.asFunc R) ∘F ((F ^opF) ×F G)) - _Represents_ : {C D : Cat} (F : Functor C D) (R : C ⊶ D) → Type _ + _Represents_ : {C D : Cat} (F : Functor C D) (R : C ⊶ D) → Type _ _Represents_ {C}{D} F R = - NatIso (Profunctor⊶.asFunc (HomProf D profF⊶[ F , Id {C = D} ])) (Profunctor⊶.asFunc R) + NatIso (Profunctor⊶.asFunc (HomProf D profF⊶[ F , Id {C = D} ])) (Profunctor⊶.asFunc R) Representable : {C D : Cat} → C ⊶ D → Type _ - Representable {C}{D} R = Σ[ F ∈ Functor C D ] (F Represents R) + Representable {C}{D} R = Σ[ F ∈ Functor C D ] (F Represents R) record Representable' {C D : Cat} (R : C ⊶ D) : Type (ℓ-max ℓ (ℓ-suc ℓ')) where module R = Profunctor⊶ R @@ -245,14 +245,14 @@ induction (r ⋆R k) ∎ - F : Functor C D + F : Functor C D F = record - { F-ob = F₀ - ; F-hom = λ f → induction ( f ⋆L unit) - ; F-id = induction (C.id ⋆L unit) ≡⟨ (λ i → induction (⋆LId unit i)) ⟩ + { F-ob = F₀ + ; F-hom = λ f → induction ( f ⋆L unit) + ; F-id = induction (C.id ⋆L unit) ≡⟨ (λ i → induction (⋆LId unit i)) ⟩ induction unit ≡⟨ sym weak-extensionality ⟩ D.id ∎ - ; F-seq = λ f g → + ; F-seq = λ f g → induction ((f ⋆C g) ⋆L unit) ≡⟨ (λ i → induction (⋆LAssoc f g unit i)) ⟩ induction (f ⋆L (g ⋆L unit)) ≡⟨ (λ i → induction (f ⋆L sym (computation (g ⋆L unit)) i)) ⟩ induction (f ⋆L (unit ⋆R (induction (g ⋆L unit)))) ≡⟨ (λ i → induction (sym (⋆L⋆RAssoc f unit (induction (g ⋆L unit))) i)) ⟩ @@ -260,17 +260,17 @@ induction (f ⋆L unit) ⋆D induction (g ⋆L unit) ∎ } - module F = Functor F + module F = Functor F unduction : ∀ {c : C.ob} {d : D.ob} → D.Hom[ F₀ c , d ] → Het[ c , d ] unduction f = unit ⋆R f - induction⁻¹ : homomorphism (HomProf D profF⊶[ F , Id ]) R + induction⁻¹ : homomorphism (HomProf D profF⊶[ F , Id ]) R induction⁻¹ = natTrans (λ x r → unduction r) λ f⋆g i r → unduction-is-natural (fst f⋆g) (snd f⋆g) r i where unduction-is-natural : ∀ {c c' d' d} → (f : C.Hom[ c , c' ])(g : D.Hom[ d' , d ])(IP : D.Hom[ F₀ c' , d' ]) - → unduction ((F ⟪ f ⟫ ⋆D IP) ⋆D g) ≡ f ⋆L⟨ unduction IP ⟩⋆R g + → unduction ((F ⟪ f ⟫ ⋆D IP) ⋆D g) ≡ f ⋆L⟨ unduction IP ⟩⋆R g unduction-is-natural f g IP = unit ⋆R ((induction (f ⋆L unit) ⋆D IP) ⋆D g) ≡⟨ (λ i → unit ⋆R D.⋆Assoc (induction (f ⋆L unit)) IP g i) ⟩ unit ⋆R (induction (f ⋆L unit) ⋆D (IP ⋆D g)) ≡⟨ ⋆RAssoc unit _ (IP ⋆D g) ⟩ @@ -304,7 +304,7 @@ Repr⇒Repr' : ∀ {C D} (R : C ⊶ D) → Representable R → Representable' R Repr⇒Repr' {C}{D} R (F , F-repr-R) = record - { F₀ = F.F-ob + { F₀ = F.F-ob ; unit = unduction D.id ; induction = induction ; computation = computation @@ -313,35 +313,35 @@ where module R = Profunctor⊶ R open R - module F = Functor F + module F = Functor F module F-repr-R = NatIso F-repr-R - induction : ∀ {c d} → Het[ c , d ] → D.Hom[ F.F-ob c , d ] + induction : ∀ {c d} → Het[ c , d ] → D.Hom[ F.F-ob c , d ] induction r = isIso.inv (NatIso.nIso F-repr-R (_ , _)) r - unduction-homo : Homomorphism (HomProf D profF⊶[ F , Id ]) R + unduction-homo : Homomorphism (HomProf D profF⊶[ F , Id ]) R unduction-homo = record { asNatTrans = F-repr-R.trans } module unduction-homo = Homomorphism unduction-homo - unduction : ∀ {c d} → D.Hom[ F.F-ob c , d ] → Het[ c , d ] + unduction : ∀ {c d} → D.Hom[ F.F-ob c , d ] → Het[ c , d ] unduction = Homomorphism.app unduction-homo computation : ∀ {c d} (r : Het[ c , d ]) → unduction D.id ⋆R induction r ≡ r computation r = (C.id ⋆L⟨ unduction D.id ⟩⋆R induction r) ≡⟨ sym (unduction-homo.homomorphism C.id D.id (induction r)) ⟩ - unduction ((F.F-hom C.id ⋆D D.id) ⋆D induction r) ≡⟨ (λ i → unduction (D.⋆IdR (F.F-hom C.id) i ⋆D induction r)) ⟩ - unduction (F.F-hom C.id ⋆D induction r) ≡⟨ (λ i → unduction (F.F-id i ⋆D induction r)) ⟩ + unduction ((F.F-hom C.id ⋆D D.id) ⋆D induction r) ≡⟨ (λ i → unduction (D.⋆IdR (F.F-hom C.id) i ⋆D induction r)) ⟩ + unduction (F.F-hom C.id ⋆D induction r) ≡⟨ (λ i → unduction (F.F-id i ⋆D induction r)) ⟩ unduction (D.id ⋆D (induction r)) ≡⟨ (λ i → unduction (D.⋆IdL (induction r) i)) ⟩ unduction (induction r) ≡⟨ (λ i → isIso.sec (NatIso.nIso F-repr-R _) i r) ⟩ r ∎ - extensionality : ∀ {c d} (f : D.Hom[ F.F-ob c , d ]) → f ≡ induction (unduction D.id ⋆R f) + extensionality : ∀ {c d} (f : D.Hom[ F.F-ob c , d ]) → f ≡ induction (unduction D.id ⋆R f) extensionality f = f ≡⟨ sym (λ i → isIso.ret (NatIso.nIso F-repr-R _) i f) ⟩ induction (unduction f) ≡⟨ (λ i → induction (unduction (sym (D.⋆IdL f) i))) ⟩ induction (unduction (D.id ⋆D f)) ≡⟨ (λ i → induction (unduction (sym (D.⋆IdL D.id) i ⋆D f))) ⟩ - induction (unduction ((D.id ⋆D D.id) ⋆D f)) ≡⟨ (λ i → induction (unduction ((sym F.F-id i ⋆D D.id) ⋆D f))) ⟩ - induction (unduction ((F.F-hom C.id ⋆D D.id) ⋆D f)) ≡⟨ (λ i → induction (unduction-homo.homomorphism C.id D.id f i)) ⟩ + induction (unduction ((D.id ⋆D D.id) ⋆D f)) ≡⟨ (λ i → induction (unduction ((sym F.F-id i ⋆D D.id) ⋆D f))) ⟩ + induction (unduction ((F.F-hom C.id ⋆D D.id) ⋆D f)) ≡⟨ (λ i → induction (unduction-homo.homomorphism C.id D.id f i)) ⟩ induction (C.id ⋆L⟨ unduction D.id ⟩⋆R f) ∎ diff --git a/Cubical.Categories.RezkCompletion.Base.html b/Cubical.Categories.RezkCompletion.Base.html index f4512b0788..1b47358773 100644 --- a/Cubical.Categories.RezkCompletion.Base.html +++ b/Cubical.Categories.RezkCompletion.Base.html @@ -30,7 +30,7 @@ -- because the universal property is naturally universal polymorphic, -- and so the predicate is not inside any universe of finite level. -isRezkCompletion : (F : Functor C D) → Typeω +isRezkCompletion : (F : Functor C D) → Typeω isRezkCompletion {D = D} F = isUnivalent D ×ω ({ℓ ℓ' : Level}{E : Category ℓ ℓ'} → isUnivalent E → isEquivalence (precomposeF E F)) @@ -39,7 +39,7 @@ open _×ω_ -makeIsRezkCompletion : {F : Functor C D} → isUnivalent D → isWeakEquivalence F → isRezkCompletion F +makeIsRezkCompletion : {F : Functor C D} → isUnivalent D → isWeakEquivalence F → isRezkCompletion F makeIsRezkCompletion univ w-equiv .fst = univ makeIsRezkCompletion univ w-equiv .snd univE = isWeakEquiv→isEquivPrecomp univE _ w-equiv \ No newline at end of file diff --git a/Cubical.Categories.RezkCompletion.Construction.html b/Cubical.Categories.RezkCompletion.Construction.html index 28f62e738f..0599d0ae9d 100644 --- a/Cubical.Categories.RezkCompletion.Construction.html +++ b/Cubical.Categories.RezkCompletion.Construction.html @@ -48,7 +48,7 @@ isUnivalentYonedaImage : isUnivalent YonedaImage isUnivalentYonedaImage = isUnivalentEssentialImage _ isUnivalentPresheafCategory - ToYonedaImage : Functor C YonedaImage + ToYonedaImage : Functor C YonedaImage ToYonedaImage = ToEssentialImage _ isWeakEquivalenceToYonedaImage : isWeakEquivalence ToYonedaImage diff --git a/Cubical.Categories.TypesOfCategories.TypeCategory.html b/Cubical.Categories.TypesOfCategories.TypeCategory.html index e1899e3f13..2e76fe6df3 100644 --- a/Cubical.Categories.TypesOfCategories.TypeCategory.html +++ b/Cubical.Categories.TypesOfCategories.TypeCategory.html @@ -60,7 +60,7 @@ module _ {ℓ ℓ' ℓ'' : Level} (C : Category ℓ ℓ') where open isTypeCategory open Category - open Functor + open Functor open NatTrans private @@ -71,7 +71,7 @@ PSTy[_] : PresheafCategory C ℓ'' .ob → Type _ PSTy[ Γ ] = Σ[ ΓA ∈ PresheafCategory C ℓ'' .ob ] Σ[ π ∈ ΓA ⇒ Γ ] - (∀ (c : C .ob) → isSurjSET {A = ΓA ⟅ c ⟆} {Γ ⟅ c ⟆} (π ⟦ c ⟧)) + (∀ (c : C .ob) → isSurjSET {A = ΓA ⟅ c ⟆} {Γ ⟅ c ⟆} (π ⟦ c ⟧)) -- just directly use types from above as context extensions PSCext : (Γ : _) → PSTy[ Γ ] → Σ[ ΓA ∈ PresheafCategory C ℓ'' .ob ] ΓA ⇒ Γ @@ -82,36 +82,36 @@ module _ {Δ Γ : PresheafCategory C ℓ'' .ob} (γ : Δ ⇒ Γ) (A'@(ΓA , π , isSurjπ) : PSTy[ Γ ]) where ΔA : PresheafCategory C ℓ'' .ob - ΔA .F-ob c = ΔATy , isSetΔA + ΔA .F-ob c = ΔATy , isSetΔA where - ΔATy = (Σ[ x ∈ fst (Δ ⟅ c ⟆) ] fiber (π ⟦ c ⟧) ((γ ⟦ c ⟧) x)) + ΔATy = (Σ[ x ∈ fst (Δ ⟅ c ⟆) ] fiber (π ⟦ c ⟧) ((γ ⟦ c ⟧) x)) isSetΔA : isSet ΔATy - isSetΔA = isOfHLevelΣ 2 (snd (Δ ⟅ c ⟆)) λ Γc → isOfHLevelΣ 2 (snd (ΓA ⟅ c ⟆)) λ ΓAc → isProp→isSet (snd (Γ ⟅ c ⟆) _ _) + isSetΔA = isOfHLevelΣ 2 (snd (Δ ⟅ c ⟆)) λ Γc → isOfHLevelΣ 2 (snd (ΓA ⟅ c ⟆)) λ ΓAc → isProp→isSet (snd (Γ ⟅ c ⟆) _ _) -- for morphisms, we apply Δ ⟪ f ⟫ to the first component -- and ΓA ⟪ f ⟫ to the second -- the fiber rule - ΔA .F-hom {c} {d} f (δax , γax , eq) - = ((Δ ⟪ f ⟫) δax) - , (((ΓA ⟪ f ⟫) γax) - , ((π ⟦ d ⟧) ((ΓA ⟪ f ⟫) γax) + ΔA .F-hom {c} {d} f (δax , γax , eq) + = ((Δ ⟪ f ⟫) δax) + , (((ΓA ⟪ f ⟫) γax) + , ((π ⟦ d ⟧) ((ΓA ⟪ f ⟫) γax) ≡[ i ]⟨ π .N-hom f i γax ⟩ - (Γ ⟪ f ⟫) ((π ⟦ c ⟧) γax) - ≡[ i ]⟨ (Γ ⟪ f ⟫) (eq i) ⟩ - (Γ ⟪ f ⟫) ((γ ⟦ c ⟧) δax) + (Γ ⟪ f ⟫) ((π ⟦ c ⟧) γax) + ≡[ i ]⟨ (Γ ⟪ f ⟫) (eq i) ⟩ + (Γ ⟪ f ⟫) ((γ ⟦ c ⟧) δax) ≡[ i ]⟨ γ .N-hom f (~ i) δax ⟩ - (γ ⟦ d ⟧) ((Δ ⟪ f ⟫) δax) + (γ ⟦ d ⟧) ((Δ ⟪ f ⟫) δax) ∎)) - ΔA .F-id {x = c} + ΔA .F-id {x = c} = funExt λ (δax , γax , eq) - → ΣPathP ((λ i → Δ .F-id i δax) - , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ - (λ i → ΓA .F-id i γax)) - ΔA .F-seq {a} {b} {c} f g + → ΣPathP ((λ i → Δ .F-id i δax) + , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ + (λ i → ΓA .F-id i γax)) + ΔA .F-seq {a} {b} {c} f g = funExt λ (δax , γax , eq) - → ΣPathP ((λ i → Δ .F-seq f g i δax) - , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ - λ i → ΓA .F-seq f g i γax) + → ΣPathP ((λ i → Δ .F-seq f g i δax) + , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ + λ i → ΓA .F-seq f g i γax) π' : ΔA ⇒ Δ π' .N-ob c (x , snd) = x π' .N-hom f = refl @@ -119,7 +119,7 @@ PSReindex = ΔA , (π' , isSurj) where - isSurj : ∀ (c : C .ob) → isSurjSET {A = ΔA ⟅ c ⟆} {B = Δ ⟅ c ⟆} (π' ⟦ c ⟧) + isSurj : ∀ (c : C .ob) → isSurjSET {A = ΔA ⟅ c ⟆} {B = Δ ⟅ c ⟆} (π' ⟦ c ⟧) isSurj c δx = (δx , isSurjπ c ((γ ⟦ c ⟧) δx)) , refl PSq : ΔA ⇒ ΓA @@ -144,11 +144,11 @@ , (λ i → (sq (~ i) ⟦ c ⟧) t)) α .N-hom {d} {c} f = funExt αHomExt where - αHomExt : ∀ (t : fst (Θ ⟅ d ⟆)) - → ((p₁ ⟦ c ⟧) ((Θ ⟪ f ⟫) t) , (p₂ ⟦ c ⟧) ((Θ ⟪ f ⟫) t), _) - ≡ ((Δ ⟪ f ⟫) ((p₁ ⟦ d ⟧) t) , (ΓA ⟪ f ⟫) ((p₂ ⟦ d ⟧) t) , _) + αHomExt : ∀ (t : fst (Θ ⟅ d ⟆)) + → ((p₁ ⟦ c ⟧) ((Θ ⟪ f ⟫) t) , (p₂ ⟦ c ⟧) ((Θ ⟪ f ⟫) t), _) + ≡ ((Δ ⟪ f ⟫) ((p₁ ⟦ d ⟧) t) , (ΓA ⟪ f ⟫) ((p₂ ⟦ d ⟧) t) , _) αHomExt t = ΣPathP ((λ i → p₁ .N-hom f i t) - , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ + , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ (λ i → p₂ .N-hom f i t)) eq : _ @@ -165,10 +165,10 @@ eqβ1 = eqβ .fst eqβ2 = eqβ .snd eqExt : ∀ (c : C .ob) - → (t : fst (Θ ⟅ c ⟆)) + → (t : fst (Θ ⟅ c ⟆)) → (α ⟦ c ⟧) t ≡ (β ⟦ c ⟧) t eqExt c t = ΣPathP ((λ i → (eqβ1 i ⟦ c ⟧) t) - , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ + , fibersEqIfRepsEq {isSetB = snd (Γ ⟅ c ⟆)} _ (λ i → (eqβ2 i ⟦ c ⟧) t)) eq≡eqβ : PathP (λ i → (p₁ ≡ (α≡β i) ●ᵛ π') @@ -177,10 +177,10 @@ , isPropNatP2 (eq .snd) (eqβ .snd) α≡β) where isPropNatP1 : isOfHLevelDep 1 (λ γ → p₁ ≡ γ ●ᵛ π') - isPropNatP1 = isOfHLevel→isOfHLevelDep 1 (λ _ → isSetNatTrans _ _) + isPropNatP1 = isOfHLevel→isOfHLevelDep 1 (λ _ → isSetNatTrans _ _) isPropNatP2 : isOfHLevelDep 1 (λ γ → p₂ ≡ γ ●ᵛ PSq) - isPropNatP2 = isOfHLevel→isOfHLevelDep 1 (λ _ → isSetNatTrans _ _) + isPropNatP2 = isOfHLevel→isOfHLevelDep 1 (λ _ → isSetNatTrans _ _) -- putting everything together isTypeCategoryPresheaf : isTypeCategory (PresheafCategory C ℓ'') diff --git a/Cubical.Categories.UnderlyingGraph.html b/Cubical.Categories.UnderlyingGraph.html index 88dfd64587..a1e578d3d6 100644 --- a/Cubical.Categories.UnderlyingGraph.html +++ b/Cubical.Categories.UnderlyingGraph.html @@ -21,7 +21,7 @@ open Category open isIso -open Functor +open Functor open NatIso hiding (sqRL; sqLL) open NatTrans @@ -31,9 +31,9 @@ Cat→Graph 𝓒 .Edge = 𝓒 .Hom[_,_] Functor→GraphHom : ∀ {ℓc ℓc' ℓd ℓd'} {𝓒 : Category ℓc ℓc'} {𝓓 : Category ℓd ℓd'} - (F : Functor 𝓒 𝓓) → GraphHom (Cat→Graph 𝓒) (Cat→Graph 𝓓) -Functor→GraphHom F ._$g_ = Functor.F-ob F -Functor→GraphHom F ._<$g>_ = Functor.F-hom F + (F : Functor 𝓒 𝓓) → GraphHom (Cat→Graph 𝓒) (Cat→Graph 𝓓) +Functor→GraphHom F ._$g_ = Functor.F-ob F +Functor→GraphHom F ._<$g>_ = Functor.F-hom F module _ (G : Graph ℓg ℓg') (𝓒 : Category ℓc ℓc') where -- Interpretation of a graph in a category @@ -43,15 +43,15 @@ {𝓒 : Category ℓc ℓc'} {𝓓 : Category ℓd ℓd'} (ı : Interp G 𝓒) - (F : Functor 𝓒 𝓓) + (F : Functor 𝓒 𝓓) → Interp G 𝓓 -(ı ⋆Interp F) ._$g_ x = Functor.F-ob F (ı $g x) -(ı ⋆Interp F) ._<$g>_ e = Functor.F-hom F (ı <$g> e) +(ı ⋆Interp F) ._$g_ x = Functor.F-ob F (ı $g x) +(ı ⋆Interp F) ._<$g>_ e = Functor.F-hom F (ı <$g> e) _∘Interp_ : ∀ {G : Graph ℓg ℓg'} {𝓒 : Category ℓc ℓc'} {𝓓 : Category ℓd ℓd'} - (F : Functor 𝓒 𝓓) + (F : Functor 𝓒 𝓓) (ı : Interp G 𝓒) → Interp G 𝓓 F ∘Interp ı = ı ⋆Interp F diff --git a/Cubical.Categories.Yoneda.html b/Cubical.Categories.Yoneda.html index 8891fca50e..f588094321 100644 --- a/Cubical.Categories.Yoneda.html +++ b/Cubical.Categories.Yoneda.html @@ -26,20 +26,20 @@ -- THE YONEDA LEMMA open NatTrans -open NatTransP -open Functor +open NatTransP +open Functor open Iso module _ {C : Category ℓ ℓ'} where open Category - yoneda : (F : Functor C (SET ℓ')) + yoneda : (F : Functor C (SET ℓ')) → (c : C .ob) - → Iso ((FUNCTOR C (SET ℓ')) [ C [ c ,-] , F ]) (fst (F ⟅ c ⟆)) + → Iso ((FUNCTOR C (SET ℓ')) [ C [ c ,-] , F ]) (fst (F ⟅ c ⟆)) yoneda F c = theIso where natType = (FUNCTOR C (SET ℓ')) [ C [ c ,-] , F ] - setType = fst (F ⟅ c ⟆) + setType = fst (F ⟅ c ⟆) -- takes a natural transformation to what it does on id ϕ : natType → setType @@ -48,19 +48,19 @@ -- takes an element x of F c and sends it to the (only) natural -- transformation which takes the identity to x Ψ : setType → natType - Ψ x .N-ob c = λ f → (F ⟪ f ⟫) x + Ψ x .N-ob c = λ f → (F ⟪ f ⟫) x Ψ x .N-hom g - = funExt (λ f → (F ⟪ f ⋆⟨ C ⟩ g ⟫) x - ≡[ i ]⟨ (F .F-seq f g) i x ⟩ - (F ⟪ g ⟫) ((F ⟪ f ⟫) x) + = funExt (λ f → (F ⟪ f ⋆⟨ C ⟩ g ⟫) x + ≡[ i ]⟨ (F .F-seq f g) i x ⟩ + (F ⟪ g ⟫) ((F ⟪ f ⟫) x) ∎) theIso : Iso natType setType theIso .fun = ϕ theIso .inv = Ψ - theIso .rightInv x i = F .F-id i x + theIso .rightInv x i = F .F-id i x theIso .leftInv α@(natTrans αo αh) = - NatTrans-≡-intro (sym αo≡βo) (symP αh≡βh) + NatTrans-≡-intro (sym αo≡βo) (symP αh≡βh) where β = Ψ (ϕ α) βo = β .N-ob @@ -75,7 +75,7 @@ ≡[ i ]⟨ αo x (C .⋆IdL f (~ i)) ⟩ αo x (C .id ⋆⟨ C ⟩ f) ≡[ i ]⟨ αh f i (C .id) ⟩ -- apply naturality - (F ⟪ f ⟫) ((αo _) (C .id)) + (F ⟪ f ⟫) ((αo _) (C .id)) ∎ -- type aliases for natural transformation @@ -89,7 +89,7 @@ isProp-hom : (ϕ : NOType) → isProp (NHType ϕ) isProp-hom γ = isPropImplicitΠ2 λ x y → isPropΠ λ f → isSetHom (SET _) - {x = (C [ c , x ]) , C .isSetHom } {F ⟅ y ⟆} _ _ + {x = (C [ c , x ]) , C .isSetHom } {F ⟅ y ⟆} _ _ isPropHomP : isOfHLevelDep 1 (λ ηo → NHType ηo) isPropHomP = isOfHLevel→isOfHLevelDep 1 λ a → isProp-hom a @@ -100,7 +100,7 @@ -- it's equivalent to apply ϕ to α then do β ⟦ c ⟧ -- or apply ϕ the the composite nat trans α ◍ β -- where ϕ takes a natural transformation to its representing element - yonedaIsNaturalInFunctor : ∀ {F G : Functor C (SET ℓ')} (c : C .ob) + yonedaIsNaturalInFunctor : ∀ {F G : Functor C (SET ℓ')} (c : C .ob) → (β : F ⇒ G) → (fun (yoneda G c) ◍ compTrans β) ≡ (β ⟦ c ⟧ ◍ fun (yoneda F c)) @@ -109,10 +109,10 @@ -- in the object -- it's equivalent to apply ϕ and then F ⟪ f ⟫ -- or to apply ϕ to the natural transformation obtained by precomposing with f - yonedaIsNaturalInOb : ∀ {F : Functor C (SET ℓ')} + yonedaIsNaturalInOb : ∀ {F : Functor C (SET ℓ')} → (c c' : C .ob) → (f : C [ c , c' ]) - → yoneda F c' .fun ◍ preComp f ≡ F ⟪ f ⟫ ◍ yoneda F c .fun + → yoneda F c' .fun ◍ preComp f ≡ F ⟪ f ⟫ ◍ yoneda F c .fun yonedaIsNaturalInOb {F = F} c c' f = funExt (λ α → (yoneda F c' .fun ◍ preComp f) α ≡⟨ refl ⟩ @@ -122,9 +122,9 @@ ≡[ i ]⟨ (α ⟦ c' ⟧) (C .⋆IdL f (~ i)) ⟩ (α ⟦ c' ⟧) (C .id ⋆⟨ C ⟩ f) ≡[ i ]⟨ (α .N-hom f i) (C .id) ⟩ - (F ⟪ f ⟫) ((α ⟦ c ⟧) (C .id)) + (F ⟪ f ⟫) ((α ⟦ c ⟧) (C .id)) ≡⟨ refl ⟩ - ((F ⟪ f ⟫) ◍ yoneda F c .fun) α + ((F ⟪ f ⟫) ◍ yoneda F c .fun) α ∎) -- Yoneda for contravariant functors @@ -132,13 +132,13 @@ open Category import Cubical.Categories.NaturalTransformation open NatTrans - yonedaᴾ : (F : Functor (C ^op) (SET ℓ')) + yonedaᴾ : (F : Functor (C ^op) (SET ℓ')) → (c : C .ob) - → Iso ((FUNCTOR (C ^op) (SET ℓ')) [ C [-, c ] , F ]) (fst (F ⟅ c ⟆)) + → Iso ((FUNCTOR (C ^op) (SET ℓ')) [ C [-, c ] , F ]) (fst (F ⟅ c ⟆)) yonedaᴾ F c = ((FUNCTOR (C ^op) (SET ℓ')) [ C [-, c ] , F ]) Iso⟨ the-iso ⟩ FUNCTOR (C ^op) (SET ℓ') [ (C ^op) [ c ,-] , F ] Iso⟨ yoneda F c ⟩ - (fst (F ⟅ c ⟆)) ∎Iso where + (fst (F ⟅ c ⟆)) ∎Iso where to : FUNCTOR (C ^op) (SET ℓ') [ C [-, c ] , F ] → FUNCTOR (C ^op) (SET ℓ') [ (C ^op) [ c ,-] , F ] @@ -153,29 +153,29 @@ the-iso = iso to fro (λ b → refl) λ a → refl -- A more universe-polymorphic Yoneda lemma -yoneda* : {C : Category ℓ ℓ'}(F : Functor C (SET ℓ'')) +yoneda* : {C : Category ℓ ℓ'}(F : Functor C (SET ℓ'')) → (c : Category.ob C) - → Iso ((FUNCTOR C (SET (ℓ-max ℓ' ℓ''))) [ LiftF {ℓ'}{ℓ''} ∘F (C [ c ,-]) - , LiftF {ℓ''}{ℓ'} ∘F F ]) - (fst (F ⟅ c ⟆)) + → Iso ((FUNCTOR C (SET (ℓ-max ℓ' ℓ''))) [ LiftF {ℓ'}{ℓ''} ∘F (C [ c ,-]) + , LiftF {ℓ''}{ℓ'} ∘F F ]) + (fst (F ⟅ c ⟆)) yoneda* {ℓ}{ℓ'}{ℓ''}{C} F c = - ((FUNCTOR C (SET (ℓ-max ℓ' ℓ''))) [ LiftF {ℓ'}{ℓ''} ∘F (C [ c ,-]) - , LiftF {ℓ''}{ℓ'} ∘F F ]) + ((FUNCTOR C (SET (ℓ-max ℓ' ℓ''))) [ LiftF {ℓ'}{ℓ''} ∘F (C [ c ,-]) + , LiftF {ℓ''}{ℓ'} ∘F F ]) Iso⟨ the-iso ⟩ ((FUNCTOR (LiftHoms C ℓ'') (SET (ℓ-max ℓ' ℓ''))) [ (LiftHoms C ℓ'' [ c ,-]) - , LiftF {ℓ''}{ℓ'} ∘F (F ∘F lowerHoms C ℓ'') ]) - Iso⟨ yoneda (LiftF {ℓ''}{ℓ'} ∘F (F ∘F lowerHoms C ℓ'')) c ⟩ - Lift {ℓ''}{ℓ'} (fst (F ⟅ c ⟆)) + , LiftF {ℓ''}{ℓ'} ∘F (F ∘F lowerHoms C ℓ'') ]) + Iso⟨ yoneda (LiftF {ℓ''}{ℓ'} ∘F (F ∘F lowerHoms C ℓ'')) c ⟩ + Lift {ℓ''}{ℓ'} (fst (F ⟅ c ⟆)) Iso⟨ invIso LiftIso ⟩ - (fst (F ⟅ c ⟆)) ∎Iso where + (fst (F ⟅ c ⟆)) ∎Iso where the-iso : Iso ((FUNCTOR C (SET (ℓ-max ℓ' ℓ''))) - [ LiftF {ℓ'}{ℓ''} ∘F (C [ c ,-]) - , LiftF {ℓ''}{ℓ'} ∘F F ]) + [ LiftF {ℓ'}{ℓ''} ∘F (C [ c ,-]) + , LiftF {ℓ''}{ℓ'} ∘F F ]) ((FUNCTOR (LiftHoms C ℓ'') (SET (ℓ-max ℓ' ℓ''))) [ (LiftHoms C ℓ'' [ c ,-]) - , LiftF {ℓ''}{ℓ'} ∘F (F ∘F lowerHoms C ℓ'') ]) + , LiftF {ℓ''}{ℓ'} ∘F (F ∘F lowerHoms C ℓ'') ]) the-iso .fun α .N-ob d f = α .N-ob d f the-iso .fun α .N-hom g = α .N-hom (g .lower) the-iso .inv β .N-ob d f = β .N-ob d f @@ -183,25 +183,25 @@ the-iso .rightInv β = refl the-iso .leftInv α = refl -yonedaᴾ* : {C : Category ℓ ℓ'}(F : Functor (C ^op) (SET ℓ'')) +yonedaᴾ* : {C : Category ℓ ℓ'}(F : Functor (C ^op) (SET ℓ'')) → (c : Category.ob C) → Iso (FUNCTOR (C ^op) (SET (ℓ-max ℓ' ℓ'')) - [ LiftF {ℓ'}{ℓ''} ∘F (C [-, c ]) - , LiftF {ℓ''}{ℓ'} ∘F F ]) - (fst (F ⟅ c ⟆)) + [ LiftF {ℓ'}{ℓ''} ∘F (C [-, c ]) + , LiftF {ℓ''}{ℓ'} ∘F F ]) + (fst (F ⟅ c ⟆)) yonedaᴾ* {ℓ}{ℓ'}{ℓ''}{C} F c = (FUNCTOR (C ^op) (SET (ℓ-max ℓ' ℓ'')) - [ LiftF {ℓ'}{ℓ''} ∘F (C [-, c ]) , LiftF {ℓ''}{ℓ'} ∘F F ]) Iso⟨ the-iso ⟩ + [ LiftF {ℓ'}{ℓ''} ∘F (C [-, c ]) , LiftF {ℓ''}{ℓ'} ∘F F ]) Iso⟨ the-iso ⟩ (FUNCTOR (C ^op) (SET (ℓ-max ℓ' ℓ'')) - [ LiftF {ℓ'}{ℓ''} ∘F ((C ^op) [ c ,-]) - , LiftF {ℓ''}{ℓ'} ∘F F ]) Iso⟨ yoneda* F c ⟩ - fst (F ⟅ c ⟆) ∎Iso where + [ LiftF {ℓ'}{ℓ''} ∘F ((C ^op) [ c ,-]) + , LiftF {ℓ''}{ℓ'} ∘F F ]) Iso⟨ yoneda* F c ⟩ + fst (F ⟅ c ⟆) ∎Iso where the-iso : Iso (FUNCTOR (C ^op) (SET (ℓ-max ℓ' ℓ'')) - [ LiftF {ℓ'}{ℓ''} ∘F (C [-, c ]) , LiftF {ℓ''}{ℓ'} ∘F F ]) + [ LiftF {ℓ'}{ℓ''} ∘F (C [-, c ]) , LiftF {ℓ''}{ℓ'} ∘F F ]) (FUNCTOR (C ^op) (SET (ℓ-max ℓ' ℓ'')) - [ LiftF {ℓ'}{ℓ''} ∘F ((C ^op) [ c ,-]) , LiftF {ℓ''}{ℓ'} ∘F F ]) + [ LiftF {ℓ'}{ℓ''} ∘F ((C ^op) [ c ,-]) , LiftF {ℓ''}{ℓ'} ∘F F ]) the-iso .fun α .N-ob = α .N-ob the-iso .fun α .N-hom = α .N-hom the-iso .inv β .N-ob = β .N-ob @@ -212,58 +212,58 @@ -- Yoneda embedding -- TODO: probably want to rename/refactor module _ {C : Category ℓ ℓ'} where - open Functor + open Functor open NatTrans open Category C - yo : ob → Functor (C ^op) (SET ℓ') + yo : ob → Functor (C ^op) (SET ℓ') yo x = C [-, x ] - YO : Functor C (PresheafCategory C ℓ') - YO .F-ob = yo - YO .F-hom f .N-ob z g = g ⋆⟨ C ⟩ f - YO .F-hom f .N-hom g i h = ⋆Assoc g h f i - YO .F-id = makeNatTransPath λ i _ → λ f → ⋆IdR f i - YO .F-seq f g = makeNatTransPath λ i _ → λ h → ⋆Assoc h f g (~ i) + YO : Functor C (PresheafCategory C ℓ') + YO .F-ob = yo + YO .F-hom f .N-ob z g = g ⋆⟨ C ⟩ f + YO .F-hom f .N-hom g i h = ⋆Assoc g h f i + YO .F-id = makeNatTransPath λ i _ → λ f → ⋆IdR f i + YO .F-seq f g = makeNatTransPath λ i _ → λ h → ⋆Assoc h f g (~ i) - module _ {x} (F : Functor (C ^op) (SET ℓ')) where - yo-yo-yo : NatTrans (yo x) F → F .F-ob x .fst + module _ {x} (F : Functor (C ^op) (SET ℓ')) where + yo-yo-yo : NatTrans (yo x) F → F .F-ob x .fst yo-yo-yo α = α .N-ob _ id - no-no-no : F .F-ob x .fst → NatTrans (yo x) F - no-no-no a .N-ob y f = F .F-hom f a - no-no-no a .N-hom f = funExt λ g i → F .F-seq g f i a + no-no-no : F .F-ob x .fst → NatTrans (yo x) F + no-no-no a .N-ob y f = F .F-hom f a + no-no-no a .N-hom f = funExt λ g i → F .F-seq g f i a - yoIso : Iso (NatTrans (yo x) F) (F .F-ob x .fst) + yoIso : Iso (NatTrans (yo x) F) (F .F-ob x .fst) yoIso .Iso.fun = yo-yo-yo yoIso .Iso.inv = no-no-no - yoIso .Iso.rightInv b i = F .F-id i b + yoIso .Iso.rightInv b i = F .F-id i b yoIso .Iso.leftInv a = makeNatTransPath (funExt λ _ → funExt rem) where - rem : ∀ {z} (x₁ : C [ z , x ]) → F .F-hom x₁ (yo-yo-yo a) ≡ (a .N-ob z) x₁ + rem : ∀ {z} (x₁ : C [ z , x ]) → F .F-hom x₁ (yo-yo-yo a) ≡ (a .N-ob z) x₁ rem g = - F .F-hom g (yo-yo-yo a) + F .F-hom g (yo-yo-yo a) ≡[ i ]⟨ a .N-hom g (~ i) id ⟩ a .N-hom g i0 id ≡[ i ]⟨ a .N-ob _ (⋆IdR g i) ⟩ (a .N-ob _) g ∎ - yoEquiv : NatTrans (yo x) F ≃ F .F-ob x .fst + yoEquiv : NatTrans (yo x) F ≃ F .F-ob x .fst yoEquiv = isoToEquiv yoIso - isFullYO : isFull YO + isFullYO : isFull YO isFullYO x y F[f] = ∣ yo-yo-yo _ F[f] , yoIso {x} (yo y) .Iso.leftInv F[f] ∣₁ - isFaithfulYO : isFaithful YO + isFaithfulYO : isFaithful YO isFaithfulYO x y f g p i = hcomp (λ j → λ{ (i = i0) → ⋆IdL f j; (i = i1) → ⋆IdL g j}) (yo-yo-yo _ (p i)) - isFullyFaithfulYO : isFullyFaithful YO + isFullyFaithfulYO : isFullyFaithful YO isFullyFaithfulYO = isFull+Faithful→isFullyFaithful {F = YO} isFullYO isFaithfulYO \ No newline at end of file diff --git a/Cubical.Codata.M.AsLimit.M.Base.html b/Cubical.Codata.M.AsLimit.M.Base.html index 78fa471feb..dd62c805f6 100644 --- a/Cubical.Codata.M.AsLimit.M.Base.html +++ b/Cubical.Codata.M.AsLimit.M.Base.html @@ -88,7 +88,7 @@ shift-iso : ∀ {ℓ} (S : Container ℓ) -> Iso (P₀ S (M S)) (M S) shift-iso S@(A , B) = P₀ S (M S) - Iso⟨ Σ-cong-iso-snd + Iso⟨ Σ-cong-iso-snd (λ x → iso (λ f → (λ n z → f z .fst n) , λ n i a → f a .snd n i) (λ (u , q) z → (λ n → u n z) , λ n i → q n i z) (λ _ → refl) @@ -148,8 +148,8 @@ (u n))) (Σ[ a ∈ A ] (Σ[ u ∈ ((n : ℕ) → B a → X (sequence S) n) ] ((n : ℕ) → π (sequence S) ∘ (u (suc n)) ≡ u n))) α-iso-step-5-Iso = - Σ-cong-iso (lemma11-Iso {S = S} (λ _ → A) (λ _ x → x)) (λ a,p → - Σ-cong-iso (pathToIso (cong (λ k → (n : ℕ) → k n) (funExt λ n → cong (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 (a,p .fst) (a,p .snd) n)))) λ u → + Σ-cong-iso (lemma11-Iso {S = S} (λ _ → A) (λ _ x → x)) (λ a,p → + Σ-cong-iso (pathToIso (cong (λ k → (n : ℕ) → k n) (funExt λ n → cong (λ k → B k → Wₙ S n) (α-iso-step-5-Iso-helper0 (a,p .fst) (a,p .snd) n)))) λ u → pathToIso (cong (λ k → (n : ℕ) → k n) (funExt λ n → α-iso-step-5-Iso-helper1-Iso (a,p .fst) (a,p .snd) u n))) α-iso-step-1-4-Iso-lem-12 : diff --git a/Cubical.Cohomology.Base.html b/Cubical.Cohomology.Base.html index 716f52d229..8d3ebe61ea 100644 --- a/Cubical.Cohomology.Base.html +++ b/Cubical.Cohomology.Base.html @@ -114,7 +114,7 @@ Cohom = CohomType k , subst AbGroupStr shiftΩTwicePath (snd (abGroupStr.π₂AbGroup k)) CohomPath : Cohom' ≡ Cohom - CohomPath = ΣPathTransport→PathΣ Cohom' Cohom (shiftΩTwicePath , refl) + CohomPath = ΣPathTransport→PathΣ Cohom' Cohom (shiftΩTwicePath , refl) CohomEquiv : AbGroupEquiv Cohom' Cohom CohomEquiv = fst (invEquiv (AbGroupPath Cohom' Cohom)) CohomPath diff --git a/Cubical.Cohomology.EilenbergMacLane.EilenbergSteenrod.html b/Cubical.Cohomology.EilenbergMacLane.EilenbergSteenrod.html index d7d3adf36b..3384cdeeb0 100644 --- a/Cubical.Cohomology.EilenbergMacLane.EilenbergSteenrod.html +++ b/Cubical.Cohomology.EilenbergMacLane.EilenbergSteenrod.html @@ -163,7 +163,7 @@ (sym (merid (pt A))) (~ i) j)) leftInv (fst (suspMapIso (negsuc zero) {A = A})) = ST.elim (λ _ → isSetPathImplicit) - λ f → cong ∣_∣₂ (Σ≡Prop (λ _ → hLevelEM G 0 _ _) + λ f → cong ∣_∣₂ (Σ≡Prop (λ _ → hLevelEM G 0 _ _) (funExt (suspToPropElim (pt A) (λ _ → hLevelEM G 0 _ _) (sym (snd f))))) @@ -180,7 +180,7 @@ ; south → refl ; (merid a i) → refl}))) snd (Suspension (satisfies-ES G)) f (negsuc zero) = - funExt λ {tt* → cong ∣_∣₂ (Σ≡Prop (λ _ → hLevelEM G 0 _ _) refl)} + funExt λ {tt* → cong ∣_∣₂ (Σ≡Prop (λ _ → hLevelEM G 0 _ _) refl)} snd (Suspension (satisfies-ES G)) f (negsuc (suc n)) = refl Exactness (satisfies-ES G) {A = A} {B = B} f (pos n) = isoToPath help where @@ -216,8 +216,8 @@ (Im (coHomHom∙ n (cfcod (fst f) , refl))) fun help (x , p) = x , To x p inv help (x , p) = x , From x p - rightInv help (x , p) = Σ≡Prop (λ _ → isPropIsInIm _ _) refl - leftInv help (x , p) = Σ≡Prop (λ _ → isPropIsInKer _ _) refl + rightInv help (x , p) = Σ≡Prop (λ _ → isPropIsInIm _ _) refl + leftInv help (x , p) = Σ≡Prop (λ _ → isPropIsInKer _ _) refl Exactness (satisfies-ES {ℓ} {ℓ'} G) {A = A} {B = B} f (negsuc n) = isoToPath help where @@ -226,8 +226,8 @@ (cfcod (fst f) , refl))) fun help (tt* , p) = tt* , ∣ tt* , refl ∣₁ inv help (tt* , p) = tt* , refl - rightInv help (tt* , p) = Σ≡Prop (λ _ → isPropIsInIm _ _) refl - leftInv help (tt* , p) = Σ≡Prop (λ _ → isPropIsInKer _ _) refl + rightInv help (tt* , p) = Σ≡Prop (λ _ → isPropIsInIm _ _) refl + leftInv help (tt* , p) = Σ≡Prop (λ _ → isPropIsInKer _ _) refl Dimension (satisfies-ES G) (pos zero) p = ⊥.rec (p refl) fst (Dimension (satisfies-ES G) (pos (suc n)) _) = 0ₕ∙ (suc n) snd (Dimension (satisfies-ES G) (pos (suc n)) _) = diff --git a/Cubical.Cohomology.EilenbergMacLane.Groups.Sn.html b/Cubical.Cohomology.EilenbergMacLane.Groups.Sn.html index 6fda7bb24e..43a0ceab73 100644 --- a/Cubical.Cohomology.EilenbergMacLane.Groups.Sn.html +++ b/Cubical.Cohomology.EilenbergMacLane.Groups.Sn.html @@ -312,7 +312,7 @@ (iso (λ f → fst f false) (λ g → (λ {true → _ ; false → g}) , refl) (λ g → refl) - λ g → Σ≡Prop (λ _ → AbGroupStr.is-set (snd G) _ _) + λ g → Σ≡Prop (λ _ → AbGroupStr.is-set (snd G) _ _) (funExt λ { false → refl ; true → sym (snd g)})) HⁿSⁿ-raw≃G G (suc n) = compEquiv diff --git a/Cubical.Cohomology.EilenbergMacLane.Gysin.html b/Cubical.Cohomology.EilenbergMacLane.Gysin.html index 0ada774816..89fedbf3ca 100644 --- a/Cubical.Cohomology.EilenbergMacLane.Gysin.html +++ b/Cubical.Cohomology.EilenbergMacLane.Gysin.html @@ -947,7 +947,7 @@ ⌣≡alt : (i : ℕ) (t : suc n ≤ i) → ⌣-hom i t ≡ alt-hom i t - ⌣≡alt i t = Σ≡Prop (λ _ → isPropIsGroupHom _ _) + ⌣≡alt i t = Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (ST.elim (λ _ → isSetPathImplicit) λ _ → refl)) diff --git a/Cubical.Cohomology.EilenbergMacLane.RingStructure.html b/Cubical.Cohomology.EilenbergMacLane.RingStructure.html index 69ed6beca2..7c56eb00b8 100644 --- a/Cubical.Cohomology.EilenbergMacLane.RingStructure.html +++ b/Cubical.Cohomology.EilenbergMacLane.RingStructure.html @@ -54,11 +54,11 @@ _⌣_ (λ {k} {l} → 0ₕ-⌣ k l) (λ {k} {l} → ⌣-0ₕ k l) - (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-assoc _ _ _) , flipTransport (assoc⌣ _ _ _ _ _ _)))) - (λ {n} a → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) + (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-assoc _ _ _) , flipTransport (assoc⌣ _ _ _ _ _ _)))) + (λ {n} a → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (⌣-1ₕ _ _ ∙ cong (λ p → subst (λ p → coHom p R-ab A) p a) (isSetℕ _ _ _ _))))) - (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ 1ₕ-⌣ _ _)) + (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ 1ₕ-⌣ _ _)) (λ _ _ _ → distrL⌣ _ _ _ _ _) λ _ _ _ → distrR⌣ _ _ _ _ _ diff --git a/Cubical.Data.Cardinality.Properties.html b/Cubical.Data.Cardinality.Properties.html index b954f14f8a..e9e088c88a 100644 --- a/Cubical.Data.Cardinality.Properties.html +++ b/Cubical.Data.Cardinality.Properties.html @@ -38,53 +38,53 @@ private +Assoc : (A B C : Card {ℓ}) → A + (B + C) ≡ (A + B) + C +Assoc = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (sym (isoToPath ⊎-assoc-Iso))) ·Assoc : (A B C : Card {ℓ}) → A · (B · C) ≡ (A · B) · C ·Assoc = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (sym (isoToPath Σ-assoc-Iso))) +IdR𝟘 : (A : Card {ℓ}) → A + 𝟘 ≡ A +IdR𝟘 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath ⊎-IdR-⊥*-Iso)) +IdL𝟘 : (A : Card {ℓ}) → 𝟘 + A ≡ A +IdL𝟘 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath ⊎-IdL-⊥*-Iso)) ·IdR𝟙 : (A : Card {ℓ}) → A · 𝟙 ≡ A ·IdR𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath rUnit*×Iso)) ·IdL𝟙 : (A : Card {ℓ}) → 𝟙 · A ≡ A ·IdL𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath lUnit*×Iso)) +Comm : (A B : Card {ℓ}) → (A + B) ≡ (B + A) +Comm = ∥₂.elim2 (λ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath ⊎-swap-Iso)) ·Comm : (A B : Card {ℓ}) → (A · B) ≡ (B · A) ·Comm = ∥₂.elim2 (λ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath Σ-swap-Iso)) ·LDist+ : (A B C : Card {ℓ}) → A · (B + C) ≡ (A · B) + (A · C) ·LDist+ = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath ×DistL⊎Iso)) AnnihilL : (A : Card {ℓ}) → 𝟘 · A ≡ 𝟘 AnnihilL = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) - (isoToPath (ΣEmpty*Iso λ _ → _))) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + (isoToPath (ΣEmpty*Iso λ _ → _))) isCardCommSemiring : IsCommSemiring {ℓ-suc ℓ} 𝟘 𝟙 _+_ _·_ isCardCommSemiring = makeIsCommSemiring isSetCard +Assoc +IdR𝟘 +Comm ·Assoc ·IdR𝟙 ·LDist+ AnnihilL ·Comm @@ -93,7 +93,7 @@ ^AnnihilR𝟘 : (A : Card {ℓ}) → A ^ 𝟘 ≡ 𝟙 ^AnnihilR𝟘 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath (iso⊥ _))) where iso⊥ : ∀ A → Iso (⊥* → A) Unit* Iso.fun (iso⊥ A) _ = tt* @@ -103,7 +103,7 @@ ^IdR𝟙 : (A : Card {ℓ}) → A ^ 𝟙 ≡ A ^IdR𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath (iso⊤ _))) where iso⊤ : ∀ A → Iso (Unit* → A) A Iso.fun (iso⊤ _) f = f tt* @@ -113,7 +113,7 @@ ^AnnihilL𝟙 : (A : Card {ℓ}) → 𝟙 ^ A ≡ 𝟙 ^AnnihilL𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) - λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath (iso⊤ _))) where iso⊤ : ∀ A → Iso (A → Unit*) Unit* Iso.fun (iso⊤ _) _ = tt* @@ -123,21 +123,21 @@ ^LDist+ : (A B C : Card {ℓ}) → A ^ (B + C) ≡ (A ^ B) · (A ^ C) ^LDist+ = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath Π⊎Iso)) ^Assoc· : (A B C : Card {ℓ}) → A ^ (B · C) ≡ (A ^ B) ^ C ^Assoc· = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath (is _ _ _))) where is : ∀ A B C → Iso (B × C → A) (C → B → A) is A B C = (B × C → A) Iso⟨ domIso Σ-swap-Iso ⟩ - (C × B → A) Iso⟨ curryIso ⟩ + (C × B → A) Iso⟨ curryIso ⟩ (C → B → A) ∎Iso ^RDist· : (A B C : Card {ℓ}) → (A · B) ^ C ≡ (A ^ C) · (B ^ C) ^RDist· = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) - λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) + λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) (isoToPath Σ-Π-Iso)) diff --git a/Cubical.Data.Fin.Arithmetic.html b/Cubical.Data.Fin.Arithmetic.html index c0db2ce207..865f1da4c2 100644 --- a/Cubical.Data.Fin.Arithmetic.html +++ b/Cubical.Data.Fin.Arithmetic.html @@ -45,7 +45,7 @@ +ₘ-assoc : {n : ℕ} (x y z : Fin (suc n)) → (x +ₘ y) +ₘ z ≡ (x +ₘ (y +ₘ z)) +ₘ-assoc {n = n} x y z = - Σ≡Prop (λ _ → isProp≤) + Σ≡Prop (λ _ → isProp≤) ((mod-rCancel (suc n) ((fst x + fst y) mod (suc n)) (fst z)) ∙∙ sym (mod+mod≡mod (suc n) (fst x + fst y) (fst z)) ∙∙ cong (_mod suc n) (sym (+-assoc (fst x) (fst y) (fst z))) @@ -54,12 +54,12 @@ +ₘ-comm : {n : ℕ} (x y : Fin (suc n)) → (x +ₘ y) ≡ (y +ₘ x) +ₘ-comm {n = n} x y = - Σ≡Prop (λ _ → isProp≤) + Σ≡Prop (λ _ → isProp≤) (cong (_mod suc n) (+-comm (fst x) (fst y))) +ₘ-lUnit : {n : ℕ} (x : Fin (suc n)) → 0 +ₘ x ≡ x +ₘ-lUnit {n = n} (x , p) = - Σ≡Prop (λ _ → isProp≤) + Σ≡Prop (λ _ → isProp≤) (+inductionBase n _ _ _ x p) +ₘ-rUnit : {n : ℕ} (x : Fin (suc n)) → x +ₘ 0 ≡ x @@ -67,7 +67,7 @@ +ₘ-rCancel : {n : ℕ} (x : Fin (suc n)) → x -ₘ x ≡ 0 +ₘ-rCancel {n = n} x = - Σ≡Prop (λ _ → isProp≤) + Σ≡Prop (λ _ → isProp≤) (cong (λ z → (fst x + z) mod (suc n)) (+inductionBase n _ _ _ (fst x) (snd x)) ∙∙ sym (mod-rCancel (suc n) (fst x) ((suc n) ∸ (fst x))) diff --git a/Cubical.Data.Fin.Base.html b/Cubical.Data.Fin.Base.html index f9abcde959..5bc21214b2 100644 --- a/Cubical.Data.Fin.Base.html +++ b/Cubical.Data.Fin.Base.html @@ -51,7 +51,7 @@ -- ... and injective. toℕ-injective : ∀{fj fk : Fin k} → toℕ fj ≡ toℕ fk → fj ≡ fk -toℕ-injective {fj = fj} {fk} = Σ≡Prop (λ _ → isProp≤) +toℕ-injective {fj = fj} {fk} = Σ≡Prop (λ _ → isProp≤) -- Conversion from ℕ with a recursive definition of ≤ @@ -109,5 +109,5 @@ FinPathℕ : {n : ℕ} (x : Fin n) (y : ℕ) → fst x ≡ y → Σ[ p ∈ _ ] (x ≡ (y , p)) FinPathℕ {n = n} x y p = ((fst (snd x)) , (cong (λ y → fst (snd x) + y) (cong suc (sym p)) ∙ snd (snd x))) - , (Σ≡Prop (λ _ → isProp≤) p) + , (Σ≡Prop (λ _ → isProp≤) p) \ No newline at end of file diff --git a/Cubical.Data.Fin.LehmerCode.html b/Cubical.Data.Fin.LehmerCode.html index 9850931fad..cf5b599356 100644 --- a/Cubical.Data.Fin.LehmerCode.html +++ b/Cubical.Data.Fin.LehmerCode.html @@ -47,7 +47,7 @@ toFinExc = fst toFinExc-injective : {i : Fin n} → {j k : FinExcept i} → toFinExc j ≡ toFinExc k → j ≡ k -toFinExc-injective = Σ≡Prop (λ _ → isPropΠ λ _ → ⊥.isProp⊥) +toFinExc-injective = Σ≡Prop (λ _ → isPropΠ λ _ → ⊥.isProp⊥) toℕExc : {i : Fin n} → FinExcept i → ℕ toℕExc = toℕ ∘ toFinExc @@ -134,17 +134,17 @@ lehmerEquiv {suc n} = (Fin (suc n) ≃ Fin (suc n)) ≃⟨ isoToEquiv i ⟩ - (Σ[ k ∈ Fin (suc n) ] (FinExcept fzero ≃ FinExcept k)) ≃⟨ Σ-cong-equiv-snd ii ⟩ - (Fin (suc n) × (Fin n ≃ Fin n)) ≃⟨ Σ-cong-equiv-snd (λ _ → lehmerEquiv) ⟩ + (Σ[ k ∈ Fin (suc n) ] (FinExcept fzero ≃ FinExcept k)) ≃⟨ Σ-cong-equiv-snd ii ⟩ + (Fin (suc n) × (Fin n ≃ Fin n)) ≃⟨ Σ-cong-equiv-snd (λ _ → lehmerEquiv) ⟩ (Fin (suc n) × LehmerCode n) ≃⟨ lehmerSucEquiv ⟩ LehmerCode (suc n) ■ where equivIn : (f : Fin (suc n) ≃ Fin (suc n)) → FinExcept fzero ≃ FinExcept (equivFun f fzero) equivIn f = FinExcept fzero - ≃⟨ Σ-cong-equiv-snd (λ _ → preCompEquiv (invEquiv (congEquiv f))) ⟩ + ≃⟨ Σ-cong-equiv-snd (λ _ → preCompEquiv (invEquiv (congEquiv f))) ⟩ (Σ[ x ∈ Fin (suc n) ] ¬ ffun fzero ≡ ffun x) - ≃⟨ Σ-cong-equiv-fst f ⟩ + ≃⟨ Σ-cong-equiv-fst f ⟩ FinExcept (ffun fzero) ■ where ffun = equivFun f @@ -199,7 +199,7 @@ lehmerFinEquiv : LehmerCode n ≃ Fin (factorial n) lehmerFinEquiv {zero} = isContr→Equiv isContrLehmerZero isContrFin1 lehmerFinEquiv {suc n} = _ ≃⟨ invEquiv lehmerSucEquiv ⟩ - _ ≃⟨ ≃-× (idEquiv _) lehmerFinEquiv ⟩ + _ ≃⟨ ≃-× (idEquiv _) lehmerFinEquiv ⟩ _ ≃⟨ factorEquiv ⟩ _ ■ diff --git a/Cubical.Data.Fin.Properties.html b/Cubical.Data.Fin.Properties.html index abcc607426..a53ed38daa 100644 --- a/Cubical.Data.Fin.Properties.html +++ b/Cubical.Data.Fin.Properties.html @@ -66,14 +66,14 @@ discreteFin : ∀ {n} → Discrete (Fin n) discreteFin {n} (x , hx) (y , hy) with discreteℕ x y -... | yes prf = yes (Σ≡Prop (λ _ → isProp≤) prf) +... | yes prf = yes (Σ≡Prop (λ _ → isProp≤) prf) ... | no prf = no λ h → prf (cong fst h) inject<-ne : ∀ {n} (i : Fin n) → ¬ inject< ≤-refl i ≡ (n , ≤-refl) inject<-ne {n} (k , k<n) p = <→≢ k<n (cong fst p) Fin-fst-≡ : ∀ {n} {i j : Fin n} → fst i ≡ fst j → i ≡ j -Fin-fst-≡ = Σ≡Prop (λ _ → isProp≤) +Fin-fst-≡ = Σ≡Prop (λ _ → isProp≤) private subst-app : (B : A → Type b) (f : (x : A) → B x) {x y : A} (x≡y : x ≡ y) → @@ -172,14 +172,14 @@ → Residue+k k n (Residue-k k n R) ≡ R Residue+k-k k n (((r , r<k) , zero) , p) = Empty.rec (<-asym r<k (lemma₀ p refl)) Residue+k-k k n ((f , suc o) , p) - = Σ≡Prop (λ tup → isSetℕ (expand× tup) (k + n)) refl + = Σ≡Prop (λ tup → isSetℕ (expand× tup) (k + n)) refl Residue-k+k : (k n : ℕ) → (R : Residue k n) → Residue-k k n (Residue+k k n R) ≡ R Residue-k+k k n ((f , o) , p) - = Σ≡Prop (λ tup → isSetℕ (expand× tup) n) refl + = Σ≡Prop (λ tup → isSetℕ (expand× tup) n) refl private Residue≃ : ∀ k n → Residue k n ≃ Residue k (k + n) @@ -385,7 +385,7 @@ g : Fin (suc n′) → Fin n′ g k = fst (f′ k) , <-trans (snd (f′ k)) m<n′ i , j , ¬q , r = pigeonhole-special g - in transport transport-prf (i , j , ¬q , Σ≡Prop (λ _ → isProp≤) (cong fst r)) + in transport transport-prf (i , j , ¬q , Σ≡Prop (λ _ → isProp≤) (cong fst r)) where n′ : ℕ n′ = k + suc m @@ -578,11 +578,11 @@ where sec-f-g : _ sec-f-g (inl (k , k<m)) with k ≤? m - sec-f-g (inl (k , k<m)) | inl _ = cong inl (Σ≡Prop (λ _ → isProp≤) refl) + sec-f-g (inl (k , k<m)) | inl _ = cong inl (Σ≡Prop (λ _ → isProp≤) refl) sec-f-g (inl (k , k<m)) | inr m≤k = Empty.rec (¬-<-and-≥ k<m m≤k) sec-f-g (inr (k , k<n)) with (m + k) ≤? m sec-f-g (inr (k , k<n)) | inl p = Empty.rec (¬m+n<m {m} {k} p) - sec-f-g (inr (k , k<n)) | inr k≥m = cong inr (Σ≡Prop (λ _ → isProp≤) rem) + sec-f-g (inr (k , k<n)) | inr k≥m = cong inr (Σ≡Prop (λ _ → isProp≤) rem) where rem : (m + k) ∸ m ≡ k rem = subst (λ - → - ∸ m ≡ k) (+-comm k m) (m+n∸n=m m k) @@ -590,8 +590,8 @@ where ret-f-g : _ ret-f-g (k , k<m+n) with k ≤? m - ret-f-g (k , k<m+n) | inl _ = Σ≡Prop (λ _ → isProp≤) refl - ret-f-g (k , k<m+n) | inr m≥k = Σ≡Prop (λ _ → isProp≤) (∸-lemma m≥k) + ret-f-g (k , k<m+n) | inl _ = Σ≡Prop (λ _ → isProp≤) refl + ret-f-g (k , k<m+n) | inr m≥k = Σ≡Prop (λ _ → isProp≤) (∸-lemma m≥k) Fin+≡Fin⊎Fin : (m n : ℕ) → Fin (m + n) ≡ Fin m ⊎ Fin n Fin+≡Fin⊎Fin m n = isoToPath (Fin+≅Fin⊎Fin m n) diff --git a/Cubical.Data.FinSet.Base.html b/Cubical.Data.FinSet.Base.html index 282939835d..37e7e74a46 100644 --- a/Cubical.Data.FinSet.Base.html +++ b/Cubical.Data.FinSet.Base.html @@ -49,7 +49,7 @@ -- isFinSet is proposition isPropIsFinSet : isProp (isFinSet A) -isPropIsFinSet p q = Σ≡PropEquiv (λ _ → isPropPropTrunc) .fst ( +isPropIsFinSet p q = Σ≡PropEquiv (λ _ → isPropPropTrunc) .fst ( Prop.elim2 (λ _ _ → isSetℕ _ _) (λ p q → Fin-inj _ _ (ua (invEquiv (SumFin≃Fin _) ⋆ (invEquiv p) ⋆ q ⋆ SumFin≃Fin _))) @@ -91,10 +91,10 @@ -- equality between finite sets/propositions FinSet≡ : (X Y : FinSet ℓ) → (X .fst ≡ Y .fst) ≃ (X ≡ Y) -FinSet≡ _ _ = Σ≡PropEquiv (λ _ → isPropIsFinSet) +FinSet≡ _ _ = Σ≡PropEquiv (λ _ → isPropIsFinSet) FinProp≡ : (X Y : FinProp ℓ) → (X .fst .fst ≡ Y .fst .fst) ≃ (X ≡ Y) -FinProp≡ X Y = compEquiv (FinSet≡ (X .fst) (Y .fst)) (Σ≡PropEquiv (λ _ → isPropIsProp)) +FinProp≡ X Y = compEquiv (FinSet≡ (X .fst) (Y .fst)) (Σ≡PropEquiv (λ _ → isPropIsProp)) -- hlevels of FinSet and FinProp diff --git a/Cubical.Data.FinSet.Binary.Large.html b/Cubical.Data.FinSet.Binary.Large.html index bd0a4d2fe6..8f372ca738 100644 --- a/Cubical.Data.FinSet.Binary.Large.html +++ b/Cubical.Data.FinSet.Binary.Large.html @@ -109,7 +109,7 @@ first = compEquiv (invEquiv P) Q Loopᴾ² : (P Q R : Bool ≃ B) → Square (Loopᴾ P Q) (Loopᴾ P R) refl (Loopᴾ Q R) - Loopᴾ² P Q R i = Σ≡Prop (λ _ → squash₁) (S i) + Loopᴾ² P Q R i = Σ≡Prop (λ _ → squash₁) (S i) where PQ : B ≃ B PQ = compEquiv (invEquiv P) Q diff --git a/Cubical.Data.FinSet.Cardinality.html b/Cubical.Data.FinSet.Cardinality.html index 41d587d736..66cd607964 100644 --- a/Cubical.Data.FinSet.Cardinality.html +++ b/Cubical.Data.FinSet.Cardinality.html @@ -203,7 +203,7 @@ sum𝟘 : sum 𝟘 f ≡ 0 sum𝟘 = isEmpty→card≡0 (_ , isFinSetΣ 𝟘 (λ x → Fin (f x) , isFinSetFin)) - ((invEquiv (Σ-cong-equiv-fst (invEquiv 𝟘≃Empty)) ⋆ ΣEmpty _) .fst) + ((invEquiv (Σ-cong-equiv-fst (invEquiv 𝟘≃Empty)) ⋆ ΣEmpty _) .fst) prod𝟘 : prod 𝟘 f ≡ 1 prod𝟘 = @@ -216,7 +216,7 @@ sum𝟙 : sum 𝟙 f ≡ f tt* sum𝟙 = cardEquiv (_ , isFinSetΣ 𝟙 (λ x → Fin (f x) , isFinSetFin)) - (Fin (f tt*) , isFinSetFin) ∣ Σ-contractFst isContrUnit* ∣₁ + (Fin (f tt*) , isFinSetFin) ∣ Σ-contractFst isContrUnit* ∣₁ prod𝟙 : prod 𝟙 f ≡ f tt* prod𝟙 = @@ -361,7 +361,7 @@ cardΣ : card (_ , isFinSetΣ X Y) ≡ sum X (λ x → card (Y x)) cardΣ = cardEquiv (_ , isFinSetΣ X Y) (_ , isFinSetΣ X (λ x → Fin (card (Y x)) , isFinSetFin)) - (Prop.map Σ-cong-equiv-snd + (Prop.map Σ-cong-equiv-snd (choice X (λ x → Y x .fst ≃ Fin (card (Y x))) (λ x → ∣≃card∣ (Y x)))) cardΠ : card (_ , isFinSetΠ X Y) ≡ prod X (λ x → card (Y x)) diff --git a/Cubical.Data.FinSet.Constructors.html b/Cubical.Data.FinSet.Constructors.html index d76b161446..1c93c3a73e 100644 --- a/Cubical.Data.FinSet.Constructors.html +++ b/Cubical.Data.FinSet.Constructors.html @@ -55,7 +55,7 @@ isFinOrd⊎ = _ , ⊎-equiv (p .snd) (q .snd) ⋆ SumFin⊎≃ _ _ isFinOrd× : isFinOrd (X × Y) - isFinOrd× = _ , Σ-cong-equiv (p .snd) (λ _ → q .snd) ⋆ SumFin×≃ _ _ + isFinOrd× = _ , Σ-cong-equiv (p .snd) (λ _ → q .snd) ⋆ SumFin×≃ _ _ module _ (X : Type ℓ )(p : isFinOrd X) @@ -69,8 +69,8 @@ isFinOrdΣ : isFinOrd (Σ X Y) isFinOrdΣ = _ , - Σ-cong-equiv {B' = λ x → Y (invEq e x)} e (transpFamily p) - ⋆ Σ-cong-equiv-snd (λ x → q (invEq e x) .snd) + Σ-cong-equiv {B' = λ x → Y (invEq e x)} e (transpFamily p) + ⋆ Σ-cong-equiv-snd (λ x → q (invEq e x) .snd) ⋆ SumFinΣ≃ _ _ isFinOrdΠ : isFinOrd ((x : X) → Y x) diff --git a/Cubical.Data.FinSet.DecidablePredicate.html b/Cubical.Data.FinSet.DecidablePredicate.html index 7323c3b8e0..2a804e8b74 100644 --- a/Cubical.Data.FinSet.DecidablePredicate.html +++ b/Cubical.Data.FinSet.DecidablePredicate.html @@ -54,15 +54,15 @@ isFinOrdSub : isFinOrd (Σ X P) isFinOrdSub = _ , - Σ-cong-equiv {B' = λ x → P (invEq e x)} e (transpFamily p) - ⋆ Σ-cong-equiv-snd (λ x → dec (invEq e x) .snd) + Σ-cong-equiv {B' = λ x → P (invEq e x)} e (transpFamily p) + ⋆ Σ-cong-equiv-snd (λ x → dec (invEq e x) .snd) ⋆ SumFinSub≃ _ (fst ∘ dec ∘ invEq e) isDecProp∃' : isDecProp ∥ Σ X P ∥₁ isDecProp∃' = _ , Prop.propTrunc≃ ( - Σ-cong-equiv {B' = λ x → P (invEq e x)} e (transpFamily p) - ⋆ Σ-cong-equiv-snd (λ x → dec (invEq e x) .snd)) + Σ-cong-equiv {B' = λ x → P (invEq e x)} e (transpFamily p) + ⋆ Σ-cong-equiv-snd (λ x → dec (invEq e x) .snd)) ⋆ SumFin∃≃ _ (fst ∘ dec ∘ invEq e) isDecProp∀' : isDecProp ((x : X) → P x) @@ -129,7 +129,7 @@ isDecProp× : isDecProp (P .fst × Q .fst) isDecProp× .fst = P .snd .fst and Q .snd .fst - isDecProp× .snd = Σ-cong-equiv (P .snd .snd) (λ _ → Q .snd .snd) ⋆ Bool→Type×≃ _ _ + isDecProp× .snd = Σ-cong-equiv (P .snd .snd) (λ _ → Q .snd .snd) ⋆ Bool→Type×≃ _ _ module _ (X : FinSet ℓ) where diff --git a/Cubical.Data.FinSet.FiniteChoice.html b/Cubical.Data.FinSet.FiniteChoice.html index 80bba73d10..90cd5daa02 100644 --- a/Cubical.Data.FinSet.FiniteChoice.html +++ b/Cubical.Data.FinSet.FiniteChoice.html @@ -38,9 +38,9 @@ ⋆ propTrunc≃ (invEquiv (isContr→≃Unit* (isContrΠ⊥ {A = Y}))) choice≃Fin {n = suc n} Y = Π⊎≃ - ⋆ Σ-cong-equiv-fst (ΠUnit (λ x → ∥ Y (inl x) ∥₁)) - ⋆ Σ-cong-equiv-snd (λ _ → choice≃Fin {n = n} (λ x → Y (inr x))) - ⋆ Σ-cong-equiv-fst (propTrunc≃ (invEquiv (ΠUnit (λ x → Y (inl x))))) + ⋆ Σ-cong-equiv-fst (ΠUnit (λ x → ∥ Y (inl x) ∥₁)) + ⋆ Σ-cong-equiv-snd (λ _ → choice≃Fin {n = n} (λ x → Y (inr x))) + ⋆ Σ-cong-equiv-fst (propTrunc≃ (invEquiv (ΠUnit (λ x → Y (inl x))))) ⋆ ∥∥-×-≃ ⋆ propTrunc≃ (invEquiv (Π⊎≃ {E = Y})) diff --git a/Cubical.Data.FinSet.Quotients.html b/Cubical.Data.FinSet.Quotients.html index 07ce6c1dc6..12e7b7a3a8 100644 --- a/Cubical.Data.FinSet.Quotients.html +++ b/Cubical.Data.FinSet.Quotients.html @@ -128,7 +128,7 @@ _∥Eff_ = Σ[ f ∈ ℙEff (X .fst) ] isEqClassEff f ∥Eff≃∥Dec : _∥Eff_ ≃ _∥Dec_ (X .fst) R (λ x x' → isDecProp→Dec (dec x x')) - ∥Eff≃∥Dec = Σ-cong-equiv (ℙEff≃ℙDec (X .fst)) isEqClassEff→isEqClass + ∥Eff≃∥Dec = Σ-cong-equiv (ℙEff≃ℙDec (X .fst)) isEqClassEff→isEqClass isFinSet∥Eff : isFinSet _∥Eff_ isFinSet∥Eff = isFinSetSub (_ , isFinSetℙEff X) (λ _ → _ , isDecPropIsEqClassEff) diff --git a/Cubical.Data.FinType.FiniteStructure.html b/Cubical.Data.FinType.FiniteStructure.html index 1e39fef6e2..5380bd873d 100644 --- a/Cubical.Data.FinType.FiniteStructure.html +++ b/Cubical.Data.FinType.FiniteStructure.html @@ -50,7 +50,7 @@ FinSetWithStrOfCard {ℓ = ℓ} S n = Σ[ X ∈ FinSetOfCard ℓ n ] S (X .fst) .fst FinSetOfCard≡ : (X Y : FinSetOfCard ℓ n) → (X .fst ≡ Y .fst) ≃ (X ≡ Y) -FinSetOfCard≡ _ _ = Σ≡PropEquiv (λ _ → isSetℕ _ _) +FinSetOfCard≡ _ _ = Σ≡PropEquiv (λ _ → isSetℕ _ _) open Iso diff --git a/Cubical.Data.FinType.Sigma.html b/Cubical.Data.FinType.Sigma.html index f0ed9bead3..dce657fe9d 100644 --- a/Cubical.Data.FinType.Sigma.html +++ b/Cubical.Data.FinType.Sigma.html @@ -67,7 +67,7 @@ isFinType0Σ : isFinType 0 (Σ (X .fst) (λ x → Y x .fst)) isFinType0Σ = isFinType0Total (Σ (X .fst) (λ x → Y x .fst)) (X .fst) (X .snd) fst - (λ x → EquivPresIsFinType 0 (fiberProjEquiv _ _ x) (Y x .snd)) + (λ x → EquivPresIsFinType 0 (fiberProjEquiv _ _ x) (Y x .snd)) -- the main result @@ -80,6 +80,6 @@ isFinType0Σ (_ , isFinTypeSuc→isFinType1 {n = suc n} (X .snd)) (λ x → _ , isFinType→isFinType0 {n = suc n} (Y x .snd)) isFinTypeΣ {n = suc n} X Y .snd a b = - EquivPresIsFinType n (ΣPathTransport≃PathΣ a b) + EquivPresIsFinType n (ΣPathTransport≃PathΣ a b) (isFinTypeΣ {n = n} (_ , X .snd .snd _ _) (λ _ → _ , Y _ .snd .snd _ _)) \ No newline at end of file diff --git a/Cubical.Data.Int.Divisibility.html b/Cubical.Data.Int.Divisibility.html index 2676e594d8..bf146d6d58 100644 --- a/Cubical.Data.Int.Divisibility.html +++ b/Cubical.Data.Int.Divisibility.html @@ -77,9 +77,9 @@ isProp∣' : isProp (m ∣' n) isProp∣' {m = pos 0} {n = n} = isSetℤ 0 n isProp∣' {m = pos (suc m)} {n = n} p q = - Σ≡Prop (λ _ → isSetℤ _ _) (·rCancel (pos (suc m)) _ _ (p .snd ∙ sym (q .snd)) (λ r → snotz (injPos r))) + Σ≡Prop (λ _ → isSetℤ _ _) (·rCancel (pos (suc m)) _ _ (p .snd ∙ sym (q .snd)) (λ r → snotz (injPos r))) isProp∣' {m = negsuc m} {n = n} p q = - Σ≡Prop (λ _ → isSetℤ _ _) (·rCancel (negsuc m) _ _ (p .snd ∙ sym (q .snd)) (negsucNotpos _ 0)) + Σ≡Prop (λ _ → isSetℤ _ _) (·rCancel (negsuc m) _ _ (p .snd ∙ sym (q .snd)) (negsucNotpos _ 0)) ∣→∣' : (m n : ℤ) → m ∣ n → m ∣' n ∣→∣' (pos 0) n ∣ c , p ∣₁ = ·Comm 0 c ∙ p diff --git a/Cubical.Data.Int.Order.html b/Cubical.Data.Int.Order.html index 0f60898297..4b86422ca9 100644 --- a/Cubical.Data.Int.Order.html +++ b/Cubical.Data.Int.Order.html @@ -45,7 +45,7 @@ isProp≤ : isProp (m ≤ n) isProp≤ {m} {n} (k , p) (l , q) - = Σ≡Prop witness-prop lemma + = Σ≡Prop witness-prop lemma where lemma : k ≡ l lemma = injPos (inj-z+ (p ∙ sym q)) diff --git a/Cubical.Data.Nat.Divisibility.html b/Cubical.Data.Nat.Divisibility.html index 2813bed035..496b747cf0 100644 --- a/Cubical.Data.Nat.Divisibility.html +++ b/Cubical.Data.Nat.Divisibility.html @@ -42,7 +42,7 @@ isProp∣' : isProp (m ∣' n) isProp∣' {zero} {n} = isSetℕ _ _ isProp∣' {suc m} {n} (c₁ , p₁) (c₂ , p₂) = - Σ≡Prop (λ _ → isSetℕ _ _) (inj-·sm {c₁} {m} {c₂} (p₁ ∙ sym p₂)) + Σ≡Prop (λ _ → isSetℕ _ _) (inj-·sm {c₁} {m} {c₂} (p₁ ∙ sym p₂)) ∣≃∣' : (m ∣ n) ≃ (m ∣' n) ∣≃∣' {zero} = propBiimpl→Equiv isProp∣ isProp∣' diff --git a/Cubical.Data.Nat.GCD.html b/Cubical.Data.Nat.GCD.html index 4cc048d72e..f6a2d15caa 100644 --- a/Cubical.Data.Nat.GCD.html +++ b/Cubical.Data.Nat.GCD.html @@ -51,7 +51,7 @@ isPropGCD : isProp (GCD m n) isPropGCD (d , dCD , gr) (d' , d'CD , gr') = - Σ≡Prop (λ _ → isPropIsGCD) (antisym∣ (gr' d dCD) (gr d' d'CD)) + Σ≡Prop (λ _ → isPropIsGCD) (antisym∣ (gr' d dCD) (gr d' d'CD)) symGCD : isGCD m n d → isGCD n m d diff --git a/Cubical.Data.Nat.Order.html b/Cubical.Data.Nat.Order.html index 24ded90c23..ea393bd683 100644 --- a/Cubical.Data.Nat.Order.html +++ b/Cubical.Data.Nat.Order.html @@ -51,7 +51,7 @@ isProp≤ : isProp (m ≤ n) isProp≤ {m} {n} (k , p) (l , q) - = Σ≡Prop witness-prop lemma + = Σ≡Prop witness-prop lemma where lemma : k ≡ l lemma = inj-+m (p ∙ (sym q)) @@ -373,7 +373,7 @@ = case dichotomy b n return (λ d → d ≡ inr (m , p)) of λ { (inl n<b) → ⊥.rec (<-asym n<b (m , +-comm m b ∙ sym p)) ; (inr (m' , q)) - → cong inr (Σ≡Prop (λ x → isSetℕ n (b + x)) (inj-m+ {m = b} (sym q ∙ p))) + → cong inr (Σ≡Prop (λ x → isSetℕ n (b + x)) (inj-m+ {m = b} (sym q ∙ p))) } b = suc b₀ diff --git a/Cubical.Data.Prod.Properties.html b/Cubical.Data.Prod.Properties.html index 5ad77f0840..4c67e5d7d8 100644 --- a/Cubical.Data.Prod.Properties.html +++ b/Cubical.Data.Prod.Properties.html @@ -5,7 +5,7 @@ open import Cubical.Core.Everything open import Cubical.Data.Prod.Base -open import Cubical.Data.Sigma renaming (_×_ to _×Σ_) hiding (prodIso ; toProdIso ; curryIso) +open import Cubical.Data.Sigma renaming (_×_ to _×Σ_) hiding (prodIso ; toProdIso ; curryIso) open import Cubical.Foundations.Prelude open import Cubical.Foundations.Equiv diff --git a/Cubical.Data.Rationals.MoreRationals.SigmaQ.Base.html b/Cubical.Data.Rationals.MoreRationals.SigmaQ.Base.html index d07c52b748..5bc30b3daa 100644 --- a/Cubical.Data.Rationals.MoreRationals.SigmaQ.Base.html +++ b/Cubical.Data.Rationals.MoreRationals.SigmaQ.Base.html @@ -33,9 +33,9 @@ []-cancelʳ : ∀ ((a , b) : ℤ × ℕ₊₁) k → [ a · pos (ℕ₊₁→ℕ k) , b ·₊₁ k ] ≡ [ a , b ] []-cancelʳ (signed s zero , b) k = - Σ≡Prop (λ _ → isPropIsGCD) (λ i → signed-zero spos s i , 1) + Σ≡Prop (λ _ → isPropIsGCD) (λ i → signed-zero spos s i , 1) []-cancelʳ (signed s (suc a) , b) k = - Σ≡Prop (λ _ → isPropIsGCD) (λ i → signedPair (·S-comm s spos i) + Σ≡Prop (λ _ → isPropIsGCD) (λ i → signedPair (·S-comm s spos i) (toCoprime-cancelʳ (suc a , b) k i)) []-cancelʳ (posneg i , b) k j = isSet→isSet' isSetℚ ([]-cancelʳ (pos zero , b) k) ([]-cancelʳ (neg zero , b) k) diff --git a/Cubical.Data.Rationals.MoreRationals.SigmaQ.Properties.html b/Cubical.Data.Rationals.MoreRationals.SigmaQ.Properties.html index aee353fc27..2364783e64 100644 --- a/Cubical.Data.Rationals.MoreRationals.SigmaQ.Properties.html +++ b/Cubical.Data.Rationals.MoreRationals.SigmaQ.Properties.html @@ -58,11 +58,11 @@ reduce-[] : ∀ x → reduce Quo.[ x .fst ] ≡ x -- equivalent to: Sigma.[ s .fst ] ≡ x reduce-[] ((signed s a , b) , cp) = - Σ≡Prop (λ _ → isPropIsGCD) (toCoprime-eq₂ s (a , b) cp) + Σ≡Prop (λ _ → isPropIsGCD) (toCoprime-eq₂ s (a , b) cp) reduce-[] ((posneg i , b) , cp) j = isSet→isSet' Sigma.isSetℚ - (Σ≡Prop (λ _ → isPropIsGCD) (toCoprime-eq₂ spos (0 , b) cp)) - (Σ≡Prop (λ _ → isPropIsGCD) (toCoprime-eq₂ sneg (0 , b) cp)) + (Σ≡Prop (λ _ → isPropIsGCD) (toCoprime-eq₂ spos (0 , b) cp)) + (Σ≡Prop (λ _ → isPropIsGCD) (toCoprime-eq₂ sneg (0 , b) cp)) (λ i → Sigma.[ posneg i , b ]) (λ i → (posneg i , b) , cp) i j diff --git a/Cubical.Data.Rationals.Order.html b/Cubical.Data.Rationals.Order.html index 92177a823b..a0b99b385f 100644 --- a/Cubical.Data.Rationals.Order.html +++ b/Cubical.Data.Rationals.Order.html @@ -69,7 +69,7 @@ toPath : ∀ a/b c/d (x : a/b ∼ c/d) (y : ℚ) → fun₀ a/b y ≡ fun₀ c/d y toPath (a , b) (c , d) ad≡cb = elimProp (λ _ → isSetHProp _ _) λ (e , f) → - Σ≡Prop (λ _ → isPropIsProp) (ua (propBiimpl→Equiv ℤ.isProp≤ ℤ.isProp≤ + Σ≡Prop (λ _ → isPropIsProp) (ua (propBiimpl→Equiv ℤ.isProp≤ ℤ.isProp≤ (ℤ.≤-·o-cancel {k = -1+ b} ∘ subst2 ℤ._≤_ (·CommR a (ℕ₊₁→ℤ f) (ℕ₊₁→ℤ d) ∙ cong (ℤ._· ℕ₊₁→ℤ f) ad≡cb ∙ @@ -125,7 +125,7 @@ toPath : ∀ a/b c/d (x : a/b ∼ c/d) (y : ℚ) → fun₀ a/b y ≡ fun₀ c/d y toPath (a , b) (c , d) ad≡cb = elimProp (λ _ → isSetHProp _ _) λ (e , f) → - Σ≡Prop (λ _ → isPropIsProp) (ua (propBiimpl→Equiv ℤ.isProp< ℤ.isProp< + Σ≡Prop (λ _ → isPropIsProp) (ua (propBiimpl→Equiv ℤ.isProp< ℤ.isProp< (ℤ.<-·o-cancel {k = -1+ b} ∘ subst2 ℤ._<_ (·CommR a (ℕ₊₁→ℤ f) (ℕ₊₁→ℤ d) ∙ cong (ℤ._· ℕ₊₁→ℤ f) ad≡cb ∙ diff --git a/Cubical.Data.Sigma.Properties.html b/Cubical.Data.Sigma.Properties.html index afdef3c9f1..1ace07eff3 100644 --- a/Cubical.Data.Sigma.Properties.html +++ b/Cubical.Data.Sigma.Properties.html @@ -181,299 +181,308 @@ unquoteDecl Σ-Π-≃ = declStrictIsoToEquiv Σ-Π-≃ Σ-Π-Iso -Σ-cong-iso-fst : (isom : Iso A A') → Iso (Σ A (B ∘ fun isom)) (Σ A' B) -fun (Σ-cong-iso-fst isom) x = fun isom (x .fst) , x .snd -inv (Σ-cong-iso-fst {B = B} isom) x = inv isom (x .fst) , subst B (sym (ε (x .fst))) (x .snd) - where - ε = isHAEquiv.rinv (snd (iso→HAEquiv isom)) -rightInv (Σ-cong-iso-fst {B = B} isom) (x , y) = ΣPathP (ε x , toPathP goal) - where - ε = isHAEquiv.rinv (snd (iso→HAEquiv isom)) - goal : subst B (ε x) (subst B (sym (ε x)) y) ≡ y - goal = sym (substComposite B (sym (ε x)) (ε x) y) - ∙∙ cong (λ x → subst B x y) (lCancel (ε x)) - ∙∙ substRefl {B = B} y -leftInv (Σ-cong-iso-fst {A = A} {B = B} isom) (x , y) = ΣPathP (leftInv isom x , toPathP goal) - where - ε = isHAEquiv.rinv (snd (iso→HAEquiv isom)) - γ = isHAEquiv.com (snd (iso→HAEquiv isom)) - - lem : (x : A) → sym (ε (fun isom x)) ∙ cong (fun isom) (leftInv isom x) ≡ refl - lem x = cong (λ a → sym (ε (fun isom x)) ∙ a) (γ x) ∙ lCancel (ε (fun isom x)) - - goal : subst B (cong (fun isom) (leftInv isom x)) (subst B (sym (ε (fun isom x))) y) ≡ y - goal = sym (substComposite B (sym (ε (fun isom x))) (cong (fun isom) (leftInv isom x)) y) - ∙∙ cong (λ a → subst B a y) (lem x) - ∙∙ substRefl {B = B} y - -Σ-cong-equiv-fst : (e : A ≃ A') → Σ A (B ∘ equivFun e) ≃ Σ A' B --- we could just do this: --- Σ-cong-equiv-fst e = isoToEquiv (Σ-cong-iso-fst (equivToIso e)) --- but the following reduces slightly better -Σ-cong-equiv-fst {A = A} {A' = A'} {B = B} e = intro , isEqIntro - where - intro : Σ A (B ∘ equivFun e) → Σ A' B - intro (a , b) = equivFun e a , b - isEqIntro : isEquiv intro - isEqIntro .equiv-proof x = ctr , isCtr where - PB : ∀ {x y} → x ≡ y → B x → B y → Type _ - PB p = PathP (λ i → B (p i)) - - open Σ x renaming (fst to a'; snd to b) - open Σ (equivCtr e a') renaming (fst to ctrA; snd to α) - ctrB : B (equivFun e ctrA) - ctrB = subst B (sym α) b - ctrP : PB α ctrB b - ctrP = symP (transport-filler (λ i → B (sym α i)) b) - ctr : fiber intro x - ctr = (ctrA , ctrB) , ΣPathP (α , ctrP) - - isCtr : ∀ y → ctr ≡ y - isCtr ((r , s) , p) = λ i → (a≡r i , b!≡s i) , ΣPathP (α≡ρ i , coh i) where - open PathPΣ p renaming (fst to ρ; snd to σ) - open PathPΣ (equivCtrPath e a' (r , ρ)) renaming (fst to a≡r; snd to α≡ρ) - - b!≡s : PB (cong (equivFun e) a≡r) ctrB s - b!≡s i = comp (λ k → B (α≡ρ i (~ k))) (λ k → (λ - { (i = i0) → ctrP (~ k) - ; (i = i1) → σ (~ k) - })) b - - coh : PathP (λ i → PB (α≡ρ i) (b!≡s i) b) ctrP σ - coh i j = fill (λ k → B (α≡ρ i (~ k))) (λ k → (λ - { (i = i0) → ctrP (~ k) - ; (i = i1) → σ (~ k) - })) (inS b) (~ j) - -Σ-cong-fst : (p : A ≡ A') → Σ A (B ∘ transport p) ≡ Σ A' B -Σ-cong-fst {B = B} p i = Σ (p i) (B ∘ transp (λ j → p (i ∨ j)) i) - -Σ-cong-iso-snd : ((x : A) → Iso (B x) (B' x)) → Iso (Σ A B) (Σ A B') -fun (Σ-cong-iso-snd isom) (x , y) = x , fun (isom x) y -inv (Σ-cong-iso-snd isom) (x , y') = x , inv (isom x) y' -rightInv (Σ-cong-iso-snd isom) (x , y) = ΣPathP (refl , rightInv (isom x) y) -leftInv (Σ-cong-iso-snd isom) (x , y') = ΣPathP (refl , leftInv (isom x) y') - -Σ-cong-equiv-snd : (∀ a → B a ≃ B' a) → Σ A B ≃ Σ A B' -Σ-cong-equiv-snd h = isoToEquiv (Σ-cong-iso-snd (equivToIso ∘ h)) - -Σ-cong-snd : ((x : A) → B x ≡ B' x) → Σ A B ≡ Σ A B' -Σ-cong-snd {A = A} p i = Σ[ x ∈ A ] (p x i) - -Σ-cong-iso : (isom : Iso A A') - → ((x : A) → Iso (B x) (B' (fun isom x))) - → Iso (Σ A B) (Σ A' B') -Σ-cong-iso isom isom' = compIso (Σ-cong-iso-snd isom') (Σ-cong-iso-fst isom) - -Σ-cong-equiv : (e : A ≃ A') - → ((x : A) → B x ≃ B' (equivFun e x)) - → Σ A B ≃ Σ A' B' -Σ-cong-equiv e e' = isoToEquiv (Σ-cong-iso (equivToIso e) (equivToIso ∘ e')) - -Σ-cong' : (p : A ≡ A') → PathP (λ i → p i → Type ℓ') B B' → Σ A B ≡ Σ A' B' -Σ-cong' p p' = cong₂ (λ (A : Type _) (B : A → Type _) → Σ A B) p p' - -Σ-cong-equiv-prop : - (e : A ≃ A') - → ((x : A ) → isProp (B x)) - → ((x : A') → isProp (B' x)) - → ((x : A) → B x → B' (equivFun e x)) - → ((x : A) → B' (equivFun e x) → B x) - → Σ A B ≃ Σ A' B' -Σ-cong-equiv-prop e prop prop' prop→ prop← = - Σ-cong-equiv e (λ x → propBiimpl→Equiv (prop x) (prop' (equivFun e x)) (prop→ x) (prop← x)) - --- Alternative version for path in Σ-types, as in the HoTT book - -ΣPathTransport : (a b : Σ A B) → Type _ -ΣPathTransport {B = B} a b = Σ[ p ∈ (fst a ≡ fst b) ] transport (λ i → B (p i)) (snd a) ≡ snd b - -IsoΣPathTransportPathΣ : (a b : Σ A B) → Iso (ΣPathTransport a b) (a ≡ b) -IsoΣPathTransportPathΣ {B = B} a b = - compIso (Σ-cong-iso-snd (λ p → invIso (PathPIsoPath (λ i → B (p i)) _ _))) - ΣPathIsoPathΣ - -ΣPathTransport≃PathΣ : (a b : Σ A B) → ΣPathTransport a b ≃ (a ≡ b) -ΣPathTransport≃PathΣ {B = B} a b = isoToEquiv (IsoΣPathTransportPathΣ a b) - -ΣPathTransport→PathΣ : (a b : Σ A B) → ΣPathTransport a b → (a ≡ b) -ΣPathTransport→PathΣ a b = Iso.fun (IsoΣPathTransportPathΣ a b) - -PathΣ→ΣPathTransport : (a b : Σ A B) → (a ≡ b) → ΣPathTransport a b -PathΣ→ΣPathTransport a b = Iso.inv (IsoΣPathTransportPathΣ a b) - -ΣPathTransport≡PathΣ : (a b : Σ A B) → ΣPathTransport a b ≡ (a ≡ b) -ΣPathTransport≡PathΣ a b = ua (ΣPathTransport≃PathΣ a b) - -Σ-contractFstIso : (c : isContr A) → Iso (Σ A B) (B (c .fst)) -fun (Σ-contractFstIso {B = B} c) p = subst B (sym (c .snd (fst p))) (snd p) -inv (Σ-contractFstIso {B = B} c) b = _ , b -rightInv (Σ-contractFstIso {B = B} c) b = - cong (λ p → subst B p b) (isProp→isSet (isContr→isProp c) _ _ _ _) ∙ transportRefl _ -fst (leftInv (Σ-contractFstIso {B = B} c) p j) = c .snd (fst p) j -snd (leftInv (Σ-contractFstIso {B = B} c) p j) = - transp (λ i → B (c .snd (fst p) (~ i ∨ j))) j (snd p) - -Σ-contractFst : (c : isContr A) → Σ A B ≃ B (c .fst) -Σ-contractFst {B = B} c = isoToEquiv (Σ-contractFstIso c) - --- a special case of the above -module _ (A : Unit → Type ℓ) where - ΣUnit : Σ Unit A ≃ A tt - unquoteDef ΣUnit = defStrictEquiv ΣUnit snd (λ { x → (tt , x) }) - -Σ-contractSnd : ((a : A) → isContr (B a)) → Σ A B ≃ A -Σ-contractSnd c = isoToEquiv isom - where - isom : Iso _ _ - isom .fun = fst - isom .inv a = a , c a .fst - isom .rightInv _ = refl - isom .leftInv (a , b) = cong (a ,_) (c a .snd b) - -isEmbeddingFstΣProp : ((x : A) → isProp (B x)) - → {u v : Σ A B} - → isEquiv (λ (p : u ≡ v) → cong fst p) -isEmbeddingFstΣProp {B = B} pB {u = u} {v = v} .equiv-proof x = ctr , isCtr - where - ctrP : u ≡ v - ctrP = ΣPathP (x , isProp→PathP (λ _ → pB _) _ _) - ctr : fiber (λ (p : u ≡ v) → cong fst p) x - ctr = ctrP , refl - - isCtr : ∀ z → ctr ≡ z - isCtr (z , p) = ΣPathP (ctrP≡ , cong (sym ∘ snd) fzsingl) where - fzsingl : Path (singl x) (x , refl) (cong fst z , sym p) - fzsingl = isContrSingl x .snd (cong fst z , sym p) - ctrSnd : SquareP (λ i j → B (fzsingl i .fst j)) (cong snd ctrP) (cong snd z) _ _ - ctrSnd = isProp→SquareP (λ _ _ → pB _) _ _ _ _ - ctrP≡ : ctrP ≡ z - ctrP≡ i = ΣPathP (fzsingl i .fst , ctrSnd i) - -Σ≡PropEquiv : ((x : A) → isProp (B x)) → {u v : Σ A B} - → (u .fst ≡ v .fst) ≃ (u ≡ v) -Σ≡PropEquiv pB = invEquiv (_ , isEmbeddingFstΣProp pB) - -Σ≡Prop : ((x : A) → isProp (B x)) → {u v : Σ A B} - → (p : u .fst ≡ v .fst) → u ≡ v -Σ≡Prop pB p = equivFun (Σ≡PropEquiv pB) p - --- dependent version -ΣPathPProp : ∀ {ℓ ℓ'} {A : I → Type ℓ} {B : (i : I) → A i → Type ℓ'} - → {u : Σ (A i0) (B i0)} {v : Σ (A i1) (B i1)} - → ((a : A (i1)) → isProp (B i1 a)) - → PathP (λ i → A i) (fst u) (fst v) - → PathP (λ i → Σ (A i) (B i)) u v -fst (ΣPathPProp {u = u} {v = v} pB p i) = p i -snd (ΣPathPProp {B = B} {u = u} {v = v} pB p i) = lem i - where - lem : PathP (λ i → B i (p i)) (snd u) (snd v) - lem = toPathP (pB _ _ _) - -≃-× : ∀ {ℓ'' ℓ'''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} {D : Type ℓ'''} → A ≃ C → B ≃ D → A × B ≃ C × D -≃-× eq1 eq2 = - map-× (fst eq1) (fst eq2) - , record - { equiv-proof - = λ {(c , d) → ((eq1⁻ c .fst .fst - , eq2⁻ d .fst .fst) - , ≡-× (eq1⁻ c .fst .snd) - (eq2⁻ d .fst .snd)) - , λ {((a , b) , p) → ΣPathP (≡-× (cong fst (eq1⁻ c .snd (a , cong fst p))) - (cong fst (eq2⁻ d .snd (b , cong snd p))) - , λ i → ≡-× (snd ((eq1⁻ c .snd (a , cong fst p)) i)) - (snd ((eq2⁻ d .snd (b , cong snd p)) i)))}}} - where - eq1⁻ = equiv-proof (eq1 .snd) - eq2⁻ = equiv-proof (eq2 .snd) - -{- Some simple ismorphisms -} - -prodIso : ∀ {ℓ ℓ' ℓ'' ℓ'''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} {D : Type ℓ'''} - → Iso A C - → Iso B D - → Iso (A × B) (C × D) -Iso.fun (prodIso iAC iBD) (a , b) = (Iso.fun iAC a) , Iso.fun iBD b -Iso.inv (prodIso iAC iBD) (c , d) = (Iso.inv iAC c) , Iso.inv iBD d -Iso.rightInv (prodIso iAC iBD) (c , d) = ΣPathP ((Iso.rightInv iAC c) , (Iso.rightInv iBD d)) -Iso.leftInv (prodIso iAC iBD) (a , b) = ΣPathP ((Iso.leftInv iAC a) , (Iso.leftInv iBD b)) - -prodEquivToIso : ∀ {ℓ'' ℓ'''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} {D : Type ℓ'''} - → (e : A ≃ C)(e' : B ≃ D) - → prodIso (equivToIso e) (equivToIso e') ≡ equivToIso (≃-× e e') -Iso.fun (prodEquivToIso e e' i) = Iso.fun (equivToIso (≃-× e e')) -Iso.inv (prodEquivToIso e e' i) = Iso.inv (equivToIso (≃-× e e')) -Iso.rightInv (prodEquivToIso e e' i) = Iso.rightInv (equivToIso (≃-× e e')) -Iso.leftInv (prodEquivToIso e e' i) = Iso.leftInv (equivToIso (≃-× e e')) - -toProdIso : {B C : A → Type ℓ} - → Iso ((a : A) → B a × C a) (((a : A) → B a) × ((a : A) → C a)) -Iso.fun toProdIso = λ f → (λ a → fst (f a)) , (λ a → snd (f a)) -Iso.inv toProdIso (f , g) = λ a → (f a) , (g a) -Iso.rightInv toProdIso (f , g) = refl -Iso.leftInv toProdIso b = refl - -module _ {A : Type ℓ} {B : A → Type ℓ'} {C : ∀ a → B a → Type ℓ''} where - curryIso : Iso (((a , b) : Σ A B) → C a b) ((a : A) → (b : B a) → C a b) - Iso.fun curryIso f a b = f (a , b) - Iso.inv curryIso f a = f (fst a) (snd a) - Iso.rightInv curryIso a = refl - Iso.leftInv curryIso f = refl - - unquoteDecl curryEquiv = declStrictIsoToEquiv curryEquiv curryIso - --- Sigma type with empty base - -module _ (A : ⊥ → Type ℓ) where - - open Iso - - ΣEmptyIso : Iso (Σ ⊥ A) ⊥ - fun ΣEmptyIso (* , _) = * - - ΣEmpty : Σ ⊥ A ≃ ⊥ - ΣEmpty = isoToEquiv ΣEmptyIso - -module _ {ℓ : Level} (A : ⊥* {ℓ} → Type ℓ) where - - open Iso - - ΣEmpty*Iso : Iso (Σ ⊥* A) ⊥* - fun ΣEmpty*Iso (* , _) = * - --- fiber of projection map - -module _ - (A : Type ℓ) - (B : A → Type ℓ') where - - private - proj : Σ A B → A - proj (a , b) = a - - module _ - (a : A) where - - open Iso - - fiberProjIso : Iso (B a) (fiber proj a) - fiberProjIso .fun b = (a , b) , refl - fiberProjIso .inv ((a' , b') , p) = subst B p b' - fiberProjIso .leftInv b i = substRefl {B = B} b i - fiberProjIso .rightInv (_ , p) i .fst .fst = p (~ i) - fiberProjIso .rightInv ((_ , b') , p) i .fst .snd = subst-filler B p b' (~ i) - fiberProjIso .rightInv (_ , p) i .snd j = p (~ i ∨ j) - - fiberProjEquiv : B a ≃ fiber proj a - fiberProjEquiv = isoToEquiv fiberProjIso - -separatedΣ : Separated A → ((a : A) → Separated (B a)) → Separated (Σ A B) -separatedΣ {B = B} sepA sepB (a , b) (a' , b') p = ΣPathTransport→PathΣ _ _ (pA , pB) - where - pA : a ≡ a' - pA = sepA a a' (λ q → p (λ r → q (cong fst r))) - - pB : subst B pA b ≡ b' - pB = sepB _ _ _ (λ q → p (λ r → q (cong (λ r' → subst B r' b) - (Separated→isSet sepA _ _ pA (cong fst r)) ∙ snd (PathΣ→ΣPathTransport _ _ r)))) +module _ {A : Type ℓ} {B : A → Type ℓ'} {B' : ∀ a → Type ℓ''} where + Σ-assoc-swap-Iso : Iso (Σ[ a ∈ Σ A B ] B' (fst a)) (Σ[ a ∈ Σ A B' ] B (fst a)) + fun Σ-assoc-swap-Iso ((x , y) , z) = ((x , z) , y) + inv Σ-assoc-swap-Iso ((x , z) , y) = ((x , y) , z) + rightInv Σ-assoc-swap-Iso _ = refl + leftInv Σ-assoc-swap-Iso _ = refl + + unquoteDecl Σ-assoc-swap-≃ = declStrictIsoToEquiv Σ-assoc-swap-≃ Σ-assoc-swap-Iso + +Σ-cong-iso-fst : (isom : Iso A A') → Iso (Σ A (B ∘ fun isom)) (Σ A' B) +fun (Σ-cong-iso-fst isom) x = fun isom (x .fst) , x .snd +inv (Σ-cong-iso-fst {B = B} isom) x = inv isom (x .fst) , subst B (sym (ε (x .fst))) (x .snd) + where + ε = isHAEquiv.rinv (snd (iso→HAEquiv isom)) +rightInv (Σ-cong-iso-fst {B = B} isom) (x , y) = ΣPathP (ε x , toPathP goal) + where + ε = isHAEquiv.rinv (snd (iso→HAEquiv isom)) + goal : subst B (ε x) (subst B (sym (ε x)) y) ≡ y + goal = sym (substComposite B (sym (ε x)) (ε x) y) + ∙∙ cong (λ x → subst B x y) (lCancel (ε x)) + ∙∙ substRefl {B = B} y +leftInv (Σ-cong-iso-fst {A = A} {B = B} isom) (x , y) = ΣPathP (leftInv isom x , toPathP goal) + where + ε = isHAEquiv.rinv (snd (iso→HAEquiv isom)) + γ = isHAEquiv.com (snd (iso→HAEquiv isom)) + + lem : (x : A) → sym (ε (fun isom x)) ∙ cong (fun isom) (leftInv isom x) ≡ refl + lem x = cong (λ a → sym (ε (fun isom x)) ∙ a) (γ x) ∙ lCancel (ε (fun isom x)) + + goal : subst B (cong (fun isom) (leftInv isom x)) (subst B (sym (ε (fun isom x))) y) ≡ y + goal = sym (substComposite B (sym (ε (fun isom x))) (cong (fun isom) (leftInv isom x)) y) + ∙∙ cong (λ a → subst B a y) (lem x) + ∙∙ substRefl {B = B} y + +Σ-cong-equiv-fst : (e : A ≃ A') → Σ A (B ∘ equivFun e) ≃ Σ A' B +-- we could just do this: +-- Σ-cong-equiv-fst e = isoToEquiv (Σ-cong-iso-fst (equivToIso e)) +-- but the following reduces slightly better +Σ-cong-equiv-fst {A = A} {A' = A'} {B = B} e = intro , isEqIntro + where + intro : Σ A (B ∘ equivFun e) → Σ A' B + intro (a , b) = equivFun e a , b + isEqIntro : isEquiv intro + isEqIntro .equiv-proof x = ctr , isCtr where + PB : ∀ {x y} → x ≡ y → B x → B y → Type _ + PB p = PathP (λ i → B (p i)) + + open Σ x renaming (fst to a'; snd to b) + open Σ (equivCtr e a') renaming (fst to ctrA; snd to α) + ctrB : B (equivFun e ctrA) + ctrB = subst B (sym α) b + ctrP : PB α ctrB b + ctrP = symP (transport-filler (λ i → B (sym α i)) b) + ctr : fiber intro x + ctr = (ctrA , ctrB) , ΣPathP (α , ctrP) + + isCtr : ∀ y → ctr ≡ y + isCtr ((r , s) , p) = λ i → (a≡r i , b!≡s i) , ΣPathP (α≡ρ i , coh i) where + open PathPΣ p renaming (fst to ρ; snd to σ) + open PathPΣ (equivCtrPath e a' (r , ρ)) renaming (fst to a≡r; snd to α≡ρ) + + b!≡s : PB (cong (equivFun e) a≡r) ctrB s + b!≡s i = comp (λ k → B (α≡ρ i (~ k))) (λ k → (λ + { (i = i0) → ctrP (~ k) + ; (i = i1) → σ (~ k) + })) b + + coh : PathP (λ i → PB (α≡ρ i) (b!≡s i) b) ctrP σ + coh i j = fill (λ k → B (α≡ρ i (~ k))) (λ k → (λ + { (i = i0) → ctrP (~ k) + ; (i = i1) → σ (~ k) + })) (inS b) (~ j) + +Σ-cong-fst : (p : A ≡ A') → Σ A (B ∘ transport p) ≡ Σ A' B +Σ-cong-fst {B = B} p i = Σ (p i) (B ∘ transp (λ j → p (i ∨ j)) i) + +Σ-cong-iso-snd : ((x : A) → Iso (B x) (B' x)) → Iso (Σ A B) (Σ A B') +fun (Σ-cong-iso-snd isom) (x , y) = x , fun (isom x) y +inv (Σ-cong-iso-snd isom) (x , y') = x , inv (isom x) y' +rightInv (Σ-cong-iso-snd isom) (x , y) = ΣPathP (refl , rightInv (isom x) y) +leftInv (Σ-cong-iso-snd isom) (x , y') = ΣPathP (refl , leftInv (isom x) y') + +Σ-cong-equiv-snd : (∀ a → B a ≃ B' a) → Σ A B ≃ Σ A B' +Σ-cong-equiv-snd h = isoToEquiv (Σ-cong-iso-snd (equivToIso ∘ h)) + +Σ-cong-snd : ((x : A) → B x ≡ B' x) → Σ A B ≡ Σ A B' +Σ-cong-snd {A = A} p i = Σ[ x ∈ A ] (p x i) + +Σ-cong-iso : (isom : Iso A A') + → ((x : A) → Iso (B x) (B' (fun isom x))) + → Iso (Σ A B) (Σ A' B') +Σ-cong-iso isom isom' = compIso (Σ-cong-iso-snd isom') (Σ-cong-iso-fst isom) + +Σ-cong-equiv : (e : A ≃ A') + → ((x : A) → B x ≃ B' (equivFun e x)) + → Σ A B ≃ Σ A' B' +Σ-cong-equiv e e' = isoToEquiv (Σ-cong-iso (equivToIso e) (equivToIso ∘ e')) + +Σ-cong' : (p : A ≡ A') → PathP (λ i → p i → Type ℓ') B B' → Σ A B ≡ Σ A' B' +Σ-cong' p p' = cong₂ (λ (A : Type _) (B : A → Type _) → Σ A B) p p' + +Σ-cong-equiv-prop : + (e : A ≃ A') + → ((x : A ) → isProp (B x)) + → ((x : A') → isProp (B' x)) + → ((x : A) → B x → B' (equivFun e x)) + → ((x : A) → B' (equivFun e x) → B x) + → Σ A B ≃ Σ A' B' +Σ-cong-equiv-prop e prop prop' prop→ prop← = + Σ-cong-equiv e (λ x → propBiimpl→Equiv (prop x) (prop' (equivFun e x)) (prop→ x) (prop← x)) + +-- Alternative version for path in Σ-types, as in the HoTT book + +ΣPathTransport : (a b : Σ A B) → Type _ +ΣPathTransport {B = B} a b = Σ[ p ∈ (fst a ≡ fst b) ] transport (λ i → B (p i)) (snd a) ≡ snd b + +IsoΣPathTransportPathΣ : (a b : Σ A B) → Iso (ΣPathTransport a b) (a ≡ b) +IsoΣPathTransportPathΣ {B = B} a b = + compIso (Σ-cong-iso-snd (λ p → invIso (PathPIsoPath (λ i → B (p i)) _ _))) + ΣPathIsoPathΣ + +ΣPathTransport≃PathΣ : (a b : Σ A B) → ΣPathTransport a b ≃ (a ≡ b) +ΣPathTransport≃PathΣ {B = B} a b = isoToEquiv (IsoΣPathTransportPathΣ a b) + +ΣPathTransport→PathΣ : (a b : Σ A B) → ΣPathTransport a b → (a ≡ b) +ΣPathTransport→PathΣ a b = Iso.fun (IsoΣPathTransportPathΣ a b) + +PathΣ→ΣPathTransport : (a b : Σ A B) → (a ≡ b) → ΣPathTransport a b +PathΣ→ΣPathTransport a b = Iso.inv (IsoΣPathTransportPathΣ a b) + +ΣPathTransport≡PathΣ : (a b : Σ A B) → ΣPathTransport a b ≡ (a ≡ b) +ΣPathTransport≡PathΣ a b = ua (ΣPathTransport≃PathΣ a b) + +Σ-contractFstIso : (c : isContr A) → Iso (Σ A B) (B (c .fst)) +fun (Σ-contractFstIso {B = B} c) p = subst B (sym (c .snd (fst p))) (snd p) +inv (Σ-contractFstIso {B = B} c) b = _ , b +rightInv (Σ-contractFstIso {B = B} c) b = + cong (λ p → subst B p b) (isProp→isSet (isContr→isProp c) _ _ _ _) ∙ transportRefl _ +fst (leftInv (Σ-contractFstIso {B = B} c) p j) = c .snd (fst p) j +snd (leftInv (Σ-contractFstIso {B = B} c) p j) = + transp (λ i → B (c .snd (fst p) (~ i ∨ j))) j (snd p) + +Σ-contractFst : (c : isContr A) → Σ A B ≃ B (c .fst) +Σ-contractFst {B = B} c = isoToEquiv (Σ-contractFstIso c) + +-- a special case of the above +module _ (A : Unit → Type ℓ) where + ΣUnit : Σ Unit A ≃ A tt + unquoteDef ΣUnit = defStrictEquiv ΣUnit snd (λ { x → (tt , x) }) + +Σ-contractSnd : ((a : A) → isContr (B a)) → Σ A B ≃ A +Σ-contractSnd c = isoToEquiv isom + where + isom : Iso _ _ + isom .fun = fst + isom .inv a = a , c a .fst + isom .rightInv _ = refl + isom .leftInv (a , b) = cong (a ,_) (c a .snd b) + +isEmbeddingFstΣProp : ((x : A) → isProp (B x)) + → {u v : Σ A B} + → isEquiv (λ (p : u ≡ v) → cong fst p) +isEmbeddingFstΣProp {B = B} pB {u = u} {v = v} .equiv-proof x = ctr , isCtr + where + ctrP : u ≡ v + ctrP = ΣPathP (x , isProp→PathP (λ _ → pB _) _ _) + ctr : fiber (λ (p : u ≡ v) → cong fst p) x + ctr = ctrP , refl + + isCtr : ∀ z → ctr ≡ z + isCtr (z , p) = ΣPathP (ctrP≡ , cong (sym ∘ snd) fzsingl) where + fzsingl : Path (singl x) (x , refl) (cong fst z , sym p) + fzsingl = isContrSingl x .snd (cong fst z , sym p) + ctrSnd : SquareP (λ i j → B (fzsingl i .fst j)) (cong snd ctrP) (cong snd z) _ _ + ctrSnd = isProp→SquareP (λ _ _ → pB _) _ _ _ _ + ctrP≡ : ctrP ≡ z + ctrP≡ i = ΣPathP (fzsingl i .fst , ctrSnd i) + +Σ≡PropEquiv : ((x : A) → isProp (B x)) → {u v : Σ A B} + → (u .fst ≡ v .fst) ≃ (u ≡ v) +Σ≡PropEquiv pB = invEquiv (_ , isEmbeddingFstΣProp pB) + +Σ≡Prop : ((x : A) → isProp (B x)) → {u v : Σ A B} + → (p : u .fst ≡ v .fst) → u ≡ v +Σ≡Prop pB p = equivFun (Σ≡PropEquiv pB) p + +-- dependent version +ΣPathPProp : ∀ {ℓ ℓ'} {A : I → Type ℓ} {B : (i : I) → A i → Type ℓ'} + → {u : Σ (A i0) (B i0)} {v : Σ (A i1) (B i1)} + → ((a : A (i1)) → isProp (B i1 a)) + → PathP (λ i → A i) (fst u) (fst v) + → PathP (λ i → Σ (A i) (B i)) u v +fst (ΣPathPProp {u = u} {v = v} pB p i) = p i +snd (ΣPathPProp {B = B} {u = u} {v = v} pB p i) = lem i + where + lem : PathP (λ i → B i (p i)) (snd u) (snd v) + lem = toPathP (pB _ _ _) + +≃-× : ∀ {ℓ'' ℓ'''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} {D : Type ℓ'''} → A ≃ C → B ≃ D → A × B ≃ C × D +≃-× eq1 eq2 = + map-× (fst eq1) (fst eq2) + , record + { equiv-proof + = λ {(c , d) → ((eq1⁻ c .fst .fst + , eq2⁻ d .fst .fst) + , ≡-× (eq1⁻ c .fst .snd) + (eq2⁻ d .fst .snd)) + , λ {((a , b) , p) → ΣPathP (≡-× (cong fst (eq1⁻ c .snd (a , cong fst p))) + (cong fst (eq2⁻ d .snd (b , cong snd p))) + , λ i → ≡-× (snd ((eq1⁻ c .snd (a , cong fst p)) i)) + (snd ((eq2⁻ d .snd (b , cong snd p)) i)))}}} + where + eq1⁻ = equiv-proof (eq1 .snd) + eq2⁻ = equiv-proof (eq2 .snd) + +{- Some simple ismorphisms -} + +prodIso : ∀ {ℓ ℓ' ℓ'' ℓ'''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} {D : Type ℓ'''} + → Iso A C + → Iso B D + → Iso (A × B) (C × D) +Iso.fun (prodIso iAC iBD) (a , b) = (Iso.fun iAC a) , Iso.fun iBD b +Iso.inv (prodIso iAC iBD) (c , d) = (Iso.inv iAC c) , Iso.inv iBD d +Iso.rightInv (prodIso iAC iBD) (c , d) = ΣPathP ((Iso.rightInv iAC c) , (Iso.rightInv iBD d)) +Iso.leftInv (prodIso iAC iBD) (a , b) = ΣPathP ((Iso.leftInv iAC a) , (Iso.leftInv iBD b)) + +prodEquivToIso : ∀ {ℓ'' ℓ'''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} {D : Type ℓ'''} + → (e : A ≃ C)(e' : B ≃ D) + → prodIso (equivToIso e) (equivToIso e') ≡ equivToIso (≃-× e e') +Iso.fun (prodEquivToIso e e' i) = Iso.fun (equivToIso (≃-× e e')) +Iso.inv (prodEquivToIso e e' i) = Iso.inv (equivToIso (≃-× e e')) +Iso.rightInv (prodEquivToIso e e' i) = Iso.rightInv (equivToIso (≃-× e e')) +Iso.leftInv (prodEquivToIso e e' i) = Iso.leftInv (equivToIso (≃-× e e')) + +toProdIso : {B C : A → Type ℓ} + → Iso ((a : A) → B a × C a) (((a : A) → B a) × ((a : A) → C a)) +Iso.fun toProdIso = λ f → (λ a → fst (f a)) , (λ a → snd (f a)) +Iso.inv toProdIso (f , g) = λ a → (f a) , (g a) +Iso.rightInv toProdIso (f , g) = refl +Iso.leftInv toProdIso b = refl + +module _ {A : Type ℓ} {B : A → Type ℓ'} {C : ∀ a → B a → Type ℓ''} where + curryIso : Iso (((a , b) : Σ A B) → C a b) ((a : A) → (b : B a) → C a b) + Iso.fun curryIso f a b = f (a , b) + Iso.inv curryIso f a = f (fst a) (snd a) + Iso.rightInv curryIso a = refl + Iso.leftInv curryIso f = refl + + unquoteDecl curryEquiv = declStrictIsoToEquiv curryEquiv curryIso + +-- Sigma type with empty base + +module _ (A : ⊥ → Type ℓ) where + + open Iso + + ΣEmptyIso : Iso (Σ ⊥ A) ⊥ + fun ΣEmptyIso (* , _) = * + + ΣEmpty : Σ ⊥ A ≃ ⊥ + ΣEmpty = isoToEquiv ΣEmptyIso + +module _ {ℓ : Level} (A : ⊥* {ℓ} → Type ℓ) where + + open Iso + + ΣEmpty*Iso : Iso (Σ ⊥* A) ⊥* + fun ΣEmpty*Iso (* , _) = * + +-- fiber of projection map + +module _ + (A : Type ℓ) + (B : A → Type ℓ') where + + private + proj : Σ A B → A + proj (a , b) = a + + module _ + (a : A) where + + open Iso + + fiberProjIso : Iso (B a) (fiber proj a) + fiberProjIso .fun b = (a , b) , refl + fiberProjIso .inv ((a' , b') , p) = subst B p b' + fiberProjIso .leftInv b i = substRefl {B = B} b i + fiberProjIso .rightInv (_ , p) i .fst .fst = p (~ i) + fiberProjIso .rightInv ((_ , b') , p) i .fst .snd = subst-filler B p b' (~ i) + fiberProjIso .rightInv (_ , p) i .snd j = p (~ i ∨ j) + + fiberProjEquiv : B a ≃ fiber proj a + fiberProjEquiv = isoToEquiv fiberProjIso + +separatedΣ : Separated A → ((a : A) → Separated (B a)) → Separated (Σ A B) +separatedΣ {B = B} sepA sepB (a , b) (a' , b') p = ΣPathTransport→PathΣ _ _ (pA , pB) + where + pA : a ≡ a' + pA = sepA a a' (λ q → p (λ r → q (cong fst r))) + + pB : subst B pA b ≡ b' + pB = sepB _ _ _ (λ q → p (λ r → q (cong (λ r' → subst B r' b) + (Separated→isSet sepA _ _ pA (cong fst r)) ∙ snd (PathΣ→ΣPathTransport _ _ r)))) \ No newline at end of file diff --git a/Cubical.Data.SumFin.Properties.html b/Cubical.Data.SumFin.Properties.html index b0064920e2..f9b2d6dd37 100644 --- a/Cubical.Data.SumFin.Properties.html +++ b/Cubical.Data.SumFin.Properties.html @@ -73,10 +73,10 @@ SumFin⊎≃ (suc m) n = ⊎-assoc-≃ ⋆ ⊎-equiv (idEquiv ⊤) (SumFin⊎≃ m n) SumFinΣ≃ : (n : ℕ)(f : Fin n → ℕ) → (Σ (Fin n) (λ x → Fin (f x))) ≃ (Fin (totalSum f)) -SumFinΣ≃ 0 f = ΣEmpty _ +SumFinΣ≃ 0 f = ΣEmpty _ SumFinΣ≃ (suc n) f = Σ⊎≃ - ⋆ ⊎-equiv (ΣUnit (λ tt → Fin (f (inl tt)))) (SumFinΣ≃ n (λ x → f (inr x))) + ⋆ ⊎-equiv (ΣUnit (λ tt → Fin (f (inl tt)))) (SumFinΣ≃ n (λ x → f (inr x))) ⋆ SumFin⊎≃ (f (inl tt)) (totalSum (λ x → f (inr x))) SumFin×≃ : (m n : ℕ) → (Fin m × Fin n) ≃ (Fin (m · n)) @@ -86,7 +86,7 @@ SumFinΠ≃ 0 _ = isContr→≃Unit (isContrΠ⊥) ⋆ invEquiv (⊎-IdR-⊥-≃) SumFinΠ≃ (suc n) f = Π⊎≃ - ⋆ Σ-cong-equiv (ΠUnit (λ tt → Fin (f (inl tt)))) (λ _ → SumFinΠ≃ n (λ x → f (inr x))) + ⋆ Σ-cong-equiv (ΠUnit (λ tt → Fin (f (inl tt)))) (λ _ → SumFinΠ≃ n (λ x → f (inr x))) ⋆ SumFin×≃ (f (inl tt)) (totalProd (λ x → f (inr x))) isNotZero : ℕ → ℕ @@ -132,7 +132,7 @@ SumFinℙ≃ 0 = isContr→≃Unit (isContrΠ⊥) ⋆ invEquiv (⊎-IdR-⊥-≃) SumFinℙ≃ (suc n) = Π⊎≃ - ⋆ Σ-cong-equiv (UnitToType≃ Bool ⋆ invEquiv SumFin2≃Bool) (λ _ → SumFinℙ≃ n) + ⋆ Σ-cong-equiv (UnitToType≃ Bool ⋆ invEquiv SumFin2≃Bool) (λ _ → SumFinℙ≃ n) ⋆ SumFin×≃ 2 (2 ^ n) -- decidable subsets of SumFin @@ -150,10 +150,10 @@ SumFinDec⊎≃ _ false = ⊎-swap-≃ ⋆ ⊎-IdR-⊥-≃ SumFinSub≃ : (n : ℕ)(f : Fin n → Bool) → Σ _ (Bool→Type ∘ f) ≃ Fin (trueCount f) -SumFinSub≃ 0 _ = ΣEmpty _ +SumFinSub≃ 0 _ = ΣEmpty _ SumFinSub≃ (suc n) f = Σ⊎≃ - ⋆ ⊎-equiv (ΣUnit (Bool→Type ∘ f ∘ inl)) (SumFinSub≃ n (f ∘ inr)) + ⋆ ⊎-equiv (ΣUnit (Bool→Type ∘ f ∘ inl)) (SumFinSub≃ n (f ∘ inr)) ⋆ SumFinDec⊎≃ _ (f (inl tt)) -- decidable quantifier @@ -167,17 +167,17 @@ trueForAll (suc n) f = f (inl tt) and trueForAll n (f ∘ inr) SumFin∃→ : (n : ℕ)(f : Fin n → Bool) → Σ _ (Bool→Type ∘ f) → Bool→Type (trueForSome n f) -SumFin∃→ 0 _ = ΣEmpty _ .fst +SumFin∃→ 0 _ = ΣEmpty _ .fst SumFin∃→ (suc n) f = Bool→Type⊎' _ _ - ∘ ⊎.map (ΣUnit (Bool→Type ∘ f ∘ inl) .fst) (SumFin∃→ n (f ∘ inr)) + ∘ ⊎.map (ΣUnit (Bool→Type ∘ f ∘ inl) .fst) (SumFin∃→ n (f ∘ inr)) ∘ Σ⊎≃ .fst SumFin∃← : (n : ℕ)(f : Fin n → Bool) → Bool→Type (trueForSome n f) → Σ _ (Bool→Type ∘ f) SumFin∃← 0 _ = ⊥.rec SumFin∃← (suc n) f = invEq Σ⊎≃ - ∘ ⊎.map (invEq (ΣUnit (Bool→Type ∘ f ∘ inl))) (SumFin∃← n (f ∘ inr)) + ∘ ⊎.map (invEq (ΣUnit (Bool→Type ∘ f ∘ inl))) (SumFin∃← n (f ∘ inr)) ∘ Bool→Type⊎ _ _ SumFin∃≃ : (n : ℕ)(f : Fin n → Bool) → ∥ Σ (Fin n) (Bool→Type ∘ f) ∥₁ ≃ Bool→Type (trueForSome n f) @@ -190,7 +190,7 @@ SumFin∀≃ 0 _ = isContr→≃Unit (isContrΠ⊥) SumFin∀≃ (suc n) f = Π⊎≃ - ⋆ Σ-cong-equiv (ΠUnit (Bool→Type ∘ f ∘ inl)) (λ _ → SumFin∀≃ n (f ∘ inr)) + ⋆ Σ-cong-equiv (ΠUnit (Bool→Type ∘ f ∘ inl)) (λ _ → SumFin∀≃ n (f ∘ inr)) ⋆ Bool→Type×≃ _ _ -- internal equality diff --git a/Cubical.Displayed.Base.html b/Cubical.Displayed.Base.html index 3c1517647b..403e9ba08f 100644 --- a/Cubical.Displayed.Base.html +++ b/Cubical.Displayed.Base.html @@ -83,7 +83,7 @@ UARel._≅_ ∫ (a , b) (a' , b') = Σ[ p ∈ a ≅ a' ] (b ≅ᴰ⟨ p ⟩ b') UARel.ua ∫ (a , b) (a' , b') = compEquiv - (Σ-cong-equiv (ua a a') (λ p → uaᴰ b p b')) + (Σ-cong-equiv (ua a a') (λ p → uaᴰ b p b')) ΣPath≃PathΣ \ No newline at end of file diff --git a/Cubical.Displayed.Prop.html b/Cubical.Displayed.Prop.html index e289c729f0..4240431bd6 100644 --- a/Cubical.Displayed.Prop.html +++ b/Cubical.Displayed.Prop.html @@ -37,7 +37,7 @@ → UARel (Σ A P) ℓ≅A 𝒮-subtype 𝒮-A propP .UARel._≅_ (a , _) (a' , _) = 𝒮-A .UARel._≅_ a a' 𝒮-subtype 𝒮-A propP .UARel.ua (a , _) (a' , _) = - compEquiv (𝒮-A .UARel.ua a a') (Σ≡PropEquiv propP) + compEquiv (𝒮-A .UARel.ua a a') (Σ≡PropEquiv propP) 𝒮ᴰ-subtype : {A : Type ℓA} {𝒮-A : UARel A ℓ≅A} {B : A → Type ℓB} (𝒮ᴰ-B : DUARel 𝒮-A B ℓ≅B) @@ -49,6 +49,6 @@ compEquiv (𝒮ᴰ-B .DUARel.uaᴰ b p b') (compEquiv - (invEquiv (Σ-contractSnd λ _ → isOfHLevelPathP' 0 (propP _ b') _ _)) + (invEquiv (Σ-contractSnd λ _ → isOfHLevelPathP' 0 (propP _ b') _ _)) ΣPath≃PathΣ) \ No newline at end of file diff --git a/Cubical.Displayed.Properties.html b/Cubical.Displayed.Properties.html index 2d39c4de31..e1059bf65f 100644 --- a/Cubical.Displayed.Properties.html +++ b/Cubical.Displayed.Properties.html @@ -139,5 +139,5 @@ -- DUARelFiberIsoOver→TotalIso produces an isomorphism of total spaces -- from a relational isomorphism between B a and (F * B) a 𝒮ᴰ-fiberIsoOver→totalIso : Iso (Σ A B) (Σ A' B') - 𝒮ᴰ-fiberIsoOver→totalIso = Σ-cong-iso F fiberIsoOver + 𝒮ᴰ-fiberIsoOver→totalIso = Σ-cong-iso F fiberIsoOver \ No newline at end of file diff --git a/Cubical.Displayed.Sigma.html b/Cubical.Displayed.Sigma.html index 4d6392ba7d..285a5208a5 100644 --- a/Cubical.Displayed.Sigma.html +++ b/Cubical.Displayed.Sigma.html @@ -69,7 +69,7 @@ Σ[ q ∈ b B.≅ᴰ⟨ p ⟩ b' ] (c C.≅ᴰ⟨ p , q ⟩ c') DUARel.uaᴰ 𝒮ᴰ-Σ (b , c) p (b' , c') = compEquiv - (Σ-cong-equiv (B.uaᴰ b p b') (λ q → C.uaᴰ c (p , q) c')) + (Σ-cong-equiv (B.uaᴰ b p b') (λ q → C.uaᴰ c (p , q) c')) ΣPath≃PathΣ -- DUARel on a non-dependent Σ-type @@ -96,7 +96,7 @@ module C = SubstRel 𝒮ˢ-C 𝒮ˢ-Σ : SubstRel 𝒮-A (λ a → Σ[ b ∈ B a ] C (a , b)) - 𝒮ˢ-Σ .act p = Σ-cong-equiv (B.act p) (λ b → C.act (p , refl)) + 𝒮ˢ-Σ .act p = Σ-cong-equiv (B.act p) (λ b → C.act (p , refl)) 𝒮ˢ-Σ .uaˢ p _ = fromPathP (DUARel.uaᴰ (𝒮ᴰ-Σ (Subst→DUA 𝒮ˢ-B) (Subst→DUA 𝒮ˢ-C)) _ p _ .fst (refl , refl)) @@ -115,6 +115,6 @@ module C = SubstRel 𝒮ˢ-C _×𝒮ˢ_ : SubstRel 𝒮-A (λ a → B a × C a) - _×𝒮ˢ_ .act p = ≃-× (B.act p) (C.act p) + _×𝒮ˢ_ .act p = ≃-× (B.act p) (C.act p) _×𝒮ˢ_ .uaˢ p (b , c) = ΣPathP (B.uaˢ p b , C.uaˢ p c) \ No newline at end of file diff --git a/Cubical.Experiments.EscardoSIP.html b/Cubical.Experiments.EscardoSIP.html index 2fbc1d68e9..74905fe721 100644 --- a/Cubical.Experiments.EscardoSIP.html +++ b/Cubical.Experiments.EscardoSIP.html @@ -32,7 +32,7 @@ Σ-≡-≃ : {X : Type ℓ} {A : X → Type ℓ'} → (σ τ : Σ X A) → ((σ ≡ τ) ≃ (Σ[ p ∈ (σ .fst) ≡ (τ .fst) ] (subst A p (σ .snd) ≡ (τ .snd)))) -Σ-≡-≃ {A = A} σ τ = invEquiv (ΣPathTransport≃PathΣ σ τ) +Σ-≡-≃ {A = A} σ τ = invEquiv (ΣPathTransport≃PathΣ σ τ) @@ -99,11 +99,11 @@ φψ : (z : (Σ Y A)) → φ (ψ z) ≡ z φψ (y , a) = - ΣPathTransport→PathΣ _ _ (η y , transportTransport⁻ (λ i → A (η y i)) a) + ΣPathTransport→PathΣ _ _ (η y , transportTransport⁻ (λ i → A (η y i)) a) -- last term proves transp (λ i → A (η y i)) i0 (transp (λ i → A (η y (~ i))) i0 a) ≡ a ψφ : (z : (Σ X (A ∘ f))) → ψ (φ z) ≡ z - ψφ (x , a) = ΣPathTransport→PathΣ _ _ (ε x , q) + ψφ (x , a) = ΣPathTransport→PathΣ _ _ (ε x , q) where b : A (f (g (f x))) b = (transp (λ i → A (η (f x) (~ i))) i0 a) diff --git a/Cubical.Experiments.Poset.html b/Cubical.Experiments.Poset.html index ab35c947ae..d95a079db9 100644 --- a/Cubical.Experiments.Poset.html +++ b/Cubical.Experiments.Poset.html @@ -94,9 +94,9 @@ NTS : (fib : fiber f p) → ((to , from) , eq) ≡ fib NTS ((φ , ψ) , eq) = - Σ≡Prop + Σ≡Prop (λ i′ → isOfHLevelSuc 2 (isSetOrder ℓ₁ X) _⊑₀_ _⊑₁_ (f i′) p) - (Σ≡Prop + (Σ≡Prop (λ _ → isPropIsOrderPreserving (X , _⊑₁_) (X , _⊑₀_) (idfun _)) (isPropIsOrderPreserving (X , _⊑₀_) (X , _⊑₁_) (idfun _) to φ)) @@ -205,7 +205,7 @@ → f ≡ g → (f , f-mono) ≡ (g , g-mono) forget-mono P Q (f , f-mono) (g , g-mono) = - Σ≡Prop (λ f → isPropΠ3 λ x y x⊑y → snd (f x ⊑[ Q ] f y)) + Σ≡Prop (λ f → isPropΠ3 λ x y x⊑y → snd (f x ⊑[ Q ] f y)) module PosetReasoning (P : Poset ℓ₀ ℓ₁) where @@ -268,7 +268,7 @@ isPosetIso-prop : (P Q : Poset ℓ₀ ℓ₁) (f : P ─m→ Q) → isProp (isPosetIso P Q f) isPosetIso-prop P Q (f , f-mono) (g₀ , sec₀ , ret₀) (g₁ , sec₁ , ret₁) = - Σ≡Prop NTS g₀=g₁ + Σ≡Prop NTS g₀=g₁ where NTS : ((g , _) : Q ─m→ P) → isProp (section f g × retract f g) NTS (g , g-mono) = isPropΣ @@ -314,10 +314,10 @@ is = iso f g sec ret sec : section to from - sec (f , _) = Σ≡Prop (isPosetIso-prop P Q) refl + sec (f , _) = Σ≡Prop (isPosetIso-prop P Q) refl ret : retract to from - ret (e , _) = Σ≡Prop (isPropIsAMonotonicEqv P Q) (Σ≡Prop isPropIsEquiv refl) + ret (e , _) = Σ≡Prop (isPropIsAMonotonicEqv P Q) (Σ≡Prop isPropIsEquiv refl) -- Once we have this equivalence, the main result is then: the type of poset -- isomorphisms between `P` and `Q` is equivalent to the type of identity proofs diff --git a/Cubical.Experiments.ZCohomologyOld.KcompPrelims.html b/Cubical.Experiments.ZCohomologyOld.KcompPrelims.html index 7855cb4d92..02ede811d6 100644 --- a/Cubical.Experiments.ZCohomologyOld.KcompPrelims.html +++ b/Cubical.Experiments.ZCohomologyOld.KcompPrelims.html @@ -182,7 +182,7 @@ sphereConnectedSpecCase = sphereConnected 3 d-mapComp : Iso (fiber d-map base) (Path (S₊ 3) north north) -d-mapComp = compIso (IsoΣPathTransportPathΣ {B = HopfSuspS¹} _ _) +d-mapComp = compIso (IsoΣPathTransportPathΣ {B = HopfSuspS¹} _ _) (congIso (invIso IsoS³TotalHopf)) is1Connected-dmap : isConnectedFun 3 d-map diff --git a/Cubical.Foundations.Cubes.HLevels.html b/Cubical.Foundations.Cubes.HLevels.html index d6f3aa91c3..b985f7d7c7 100644 --- a/Cubical.Foundations.Cubes.HLevels.html +++ b/Cubical.Foundations.Cubes.HLevels.html @@ -50,9 +50,9 @@ isCubeFilledPath≡Suc n A = (λ i → (x y : A)(∂ : ∂CubeConst₀₁≡∂CubePath {n = suc n} {a₀ = x} {y} (~ i)) → CubeRelConst₀₁≡CubeRelPath (~ i) ∂) - ∙ (λ i → (x : A) → isoToPath (curryIso {A = A} + ∙ (λ i → (x : A) → isoToPath (curryIso {A = A} {B = λ y → ∂CubeConst₀₁ (suc n) A x y} {C = λ _ ∂ → CubeRelConst₀₁ (suc n) A ∂}) (~ i)) - ∙ sym (isoToPath curryIso) + ∙ sym (isoToPath curryIso) ∙ (λ i → (∂ : ∂CubeConst₀₁≡∂CubeSuc {A = A} i) → CubeRelConst₀₁≡CubeRelSuc {n = n} i ∂) isCubeFilledPath→Suc : (n : ℕ) (A : Type ℓ) diff --git a/Cubical.Foundations.Cubes.Subtypes.html b/Cubical.Foundations.Cubes.Subtypes.html index 66568197aa..c13cfb2966 100644 --- a/Cubical.Foundations.Cubes.Subtypes.html +++ b/Cubical.Foundations.Cubes.Subtypes.html @@ -203,7 +203,7 @@ -- For simplicity, we begin at n=1, and that is all we need. ∂CubeConst₀₁≃∂CubeSuc : {n : ℕ} {A : Type ℓ} → Σ∂CubeConst₀₁ (suc n) A ≃ ∂Cube (suc (suc n)) A -∂CubeConst₀₁≃∂CubeSuc = Σ-cong-equiv (_ , isEquivConst) (λ a → Σ-cong-equiv-fst (_ , isEquivConst)) +∂CubeConst₀₁≃∂CubeSuc = Σ-cong-equiv (_ , isEquivConst) (λ a → Σ-cong-equiv-fst (_ , isEquivConst)) HAEquiv-∂CubeConst₀₁-∂CubeSuc : {n : ℕ}{A : Type ℓ} → HAEquiv (Σ∂CubeConst₀₁ (suc n) A) (∂Cube (suc (suc n)) A) HAEquiv-∂CubeConst₀₁-∂CubeSuc = equiv→HAEquiv ∂CubeConst₀₁≃∂CubeSuc diff --git a/Cubical.Foundations.Equiv.Properties.html b/Cubical.Foundations.Equiv.Properties.html index c03a336418..8225c966fa 100644 --- a/Cubical.Foundations.Equiv.Properties.html +++ b/Cubical.Foundations.Equiv.Properties.html @@ -194,13 +194,13 @@ (λ _ → refl) λ _ → refl) ⟩ Σ _ rCoh1 -- secondly, convert the path into a dependent path for later convenience - ≃⟨ Σ-cong-equiv-snd (λ s → Σ-cong-equiv-snd + ≃⟨ Σ-cong-equiv-snd (λ s → Σ-cong-equiv-snd λ η → equivΠCod λ x → compEquiv (flipSquareEquiv {a₀₀ = f x}) (invEquiv slideSquareEquiv)) ⟩ Σ _ rCoh2 - ≃⟨ Σ-cong-equiv-snd (λ s → invEquiv Σ-Π-≃) ⟩ + ≃⟨ Σ-cong-equiv-snd (λ s → invEquiv Σ-Π-≃) ⟩ Σ _ rCoh3 - ≃⟨ Σ-cong-equiv-snd (λ s → equivΠCod λ x → ΣPath≃PathΣ) ⟩ + ≃⟨ Σ-cong-equiv-snd (λ s → equivΠCod λ x → ΣPath≃PathΣ) ⟩ Σ _ rCoh4 ■ where open isHAEquiv diff --git a/Cubical.Foundations.HLevels.html b/Cubical.Foundations.HLevels.html index ae857ccd32..411eb21537 100644 --- a/Cubical.Foundations.HLevels.html +++ b/Cubical.Foundations.HLevels.html @@ -135,7 +135,7 @@ isPropIs2Groupoid = isPropIsOfHLevel 4 TypeOfHLevel≡ : (n : HLevel) {X Y : TypeOfHLevel ℓ n} → ⟨ X ⟩ ≡ ⟨ Y ⟩ → X ≡ Y -TypeOfHLevel≡ n = Σ≡Prop (λ _ → isPropIsOfHLevel n) +TypeOfHLevel≡ n = Σ≡Prop (λ _ → isPropIsOfHLevel n) -- hlevels are preserved by retracts (and consequently equivalences) @@ -314,20 +314,20 @@ section-Σ≡Prop : (pB : (x : A) → isProp (B x)) {u v : Σ A B} - → section (Σ≡Prop pB {u} {v}) (cong fst) + → section (Σ≡Prop pB {u} {v}) (cong fst) section-Σ≡Prop {A = A} pB {u} {v} p j i = (p i .fst) , isProp→PathP (λ i → isOfHLevelPath 1 (pB (fst (p i))) - (Σ≡Prop pB {u} {v} (cong fst p) i .snd) + (Σ≡Prop pB {u} {v} (cong fst p) i .snd) (p i .snd) ) refl refl i j isEquiv-Σ≡Prop : (pB : (x : A) → isProp (B x)) {u v : Σ A B} - → isEquiv (Σ≡Prop pB {u} {v}) -isEquiv-Σ≡Prop {A = A} pB {u} {v} = isoToIsEquiv (iso (Σ≡Prop pB) (cong fst) (section-Σ≡Prop pB) (λ _ → refl)) + → isEquiv (Σ≡Prop pB {u} {v}) +isEquiv-Σ≡Prop {A = A} pB {u} {v} = isoToIsEquiv (iso (Σ≡Prop pB) (cong fst) (section-Σ≡Prop pB) (λ _ → refl)) isPropΣ : isProp A → ((x : A) → isProp (B x)) → isProp (Σ A B) -isPropΣ pA pB t u = Σ≡Prop pB (pA (t .fst) (u .fst)) +isPropΣ pA pB t u = Σ≡Prop pB (pA (t .fst) (u .fst)) isOfHLevelΣ : ∀ n → isOfHLevel n A → ((x : A) → isOfHLevel n (B x)) → isOfHLevel n (Σ A B) @@ -335,7 +335,7 @@ isOfHLevelΣ 1 = isPropΣ isOfHLevelΣ {B = B} (suc (suc n)) h1 h2 x y = isOfHLevelRetractFromIso (suc n) - (invIso (IsoΣPathTransportPathΣ _ _)) + (invIso (IsoΣPathTransportPathΣ _ _)) (isOfHLevelΣ (suc n) (h1 (fst x) (fst y)) λ x → h2 _ _ _) isSetΣ : isSet A → ((x : A) → isSet (B x)) → isSet (Σ A B) @@ -507,7 +507,7 @@ isOfHLevel≃ zero {A = A} {B = B} hA hB = isContr→Equiv hA hB , contr where contr : (y : A ≃ B) → isContr→Equiv hA hB ≡ y - contr y = Σ≡Prop isPropIsEquiv (funExt (λ a → snd hB (fst y a))) + contr y = Σ≡Prop isPropIsEquiv (funExt (λ a → snd hB (fst y a))) isOfHLevel≃ (suc n) {A = A} {B = B} hA hB = isOfHLevelΣ (suc n) (isOfHLevelΠ _ λ _ → hB) @@ -556,12 +556,12 @@ -- h-level of TypeOfHLevel isPropHContr : isProp (TypeOfHLevel ℓ 0) -isPropHContr x y = Σ≡Prop (λ _ → isPropIsContr) (isOfHLevel≡ 0 (x .snd) (y .snd) .fst) +isPropHContr x y = Σ≡Prop (λ _ → isPropIsContr) (isOfHLevel≡ 0 (x .snd) (y .snd) .fst) isOfHLevelTypeOfHLevel : ∀ n → isOfHLevel (suc n) (TypeOfHLevel ℓ n) isOfHLevelTypeOfHLevel zero = isPropHContr isOfHLevelTypeOfHLevel (suc n) (X , a) (Y , b) = - isOfHLevelRetract (suc n) (cong fst) (Σ≡Prop λ _ → isPropIsOfHLevel (suc n)) + isOfHLevelRetract (suc n) (cong fst) (Σ≡Prop λ _ → isPropIsOfHLevel (suc n)) (section-Σ≡Prop λ _ → isPropIsOfHLevel (suc n)) (isOfHLevel≡ (suc n) a b) diff --git a/Cubical.Foundations.Pointed.Homotopy.html b/Cubical.Foundations.Pointed.Homotopy.html index 292de6d206..5fd82f5835 100644 --- a/Cubical.Foundations.Pointed.Homotopy.html +++ b/Cubical.Foundations.Pointed.Homotopy.html @@ -95,7 +95,7 @@ -- Proof that ∙∼ and ∙∼P are equivalent using the fiberwise equivalence φ ∙∼≃∙∼P : (f g : Π∙ A B ptB) → (f ∙∼ g) ≃ (f ∙∼P g) - ∙∼≃∙∼P f g = Σ-cong-equiv-snd (λ H → pathToEquiv (P≡Q f g H)) + ∙∼≃∙∼P f g = Σ-cong-equiv-snd (λ H → pathToEquiv (P≡Q f g H)) -- inverse of ∙∼→∙∼P extracted from the equivalence ∙∼P→∙∼ : {f g : Π∙ A B ptB} → f ∙∼P g → f ∙∼ g diff --git a/Cubical.Foundations.SIP.html b/Cubical.Foundations.SIP.html index bd019a2a4b..8b2d24a1e7 100644 --- a/Cubical.Foundations.SIP.html +++ b/Cubical.Foundations.SIP.html @@ -119,7 +119,7 @@ SIP : A ≃[ ι ] B ≃ (A ≡ B) SIP = - sip , isoToIsEquiv (compIso (Σ-cong-iso (invIso univalenceIso) (equivToIso ∘ θ)) ΣPathIsoPathΣ) + sip , isoToIsEquiv (compIso (Σ-cong-iso (invIso univalenceIso) (equivToIso ∘ θ)) ΣPathIsoPathΣ) sip⁻ : A ≡ B → A ≃[ ι ] B sip⁻ = invEq SIP diff --git a/Cubical.Functions.Embedding.html b/Cubical.Functions.Embedding.html index 0b1480d869..e82e0042ea 100644 --- a/Cubical.Functions.Embedding.html +++ b/Cubical.Functions.Embedding.html @@ -160,7 +160,7 @@ injective→hasPropFibers : hasPropFibers f injective→hasPropFibers y (x , fx≡y) (x' , fx'≡y) = - Σ≡Prop + Σ≡Prop (λ _ → isSetB _ _) (inj (fx≡y ∙ sym (fx'≡y))) @@ -243,14 +243,14 @@ D = Σ[ x ∈ X ] x ∈ A Ψ : isEmbedding fst - Ψ w x = isEmbeddingFstΣProp (∈-isProp A) + Ψ w x = isEmbeddingFstΣProp (∈-isProp A) Subset→Embedding→Subset : {X : Type ℓ} → section (Embedding→Subset {ℓ} {X}) (Subset→Embedding {ℓ} {X}) -Subset→Embedding→Subset _ = funExt λ x → Σ≡Prop (λ _ → isPropIsProp) (ua (FiberIso.fiberEquiv _ x)) +Subset→Embedding→Subset _ = funExt λ x → Σ≡Prop (λ _ → isPropIsProp) (ua (FiberIso.fiberEquiv _ x)) Embedding→Subset→Embedding : {X : Type ℓ} → retract (Embedding→Subset {ℓ} {X}) (Subset→Embedding {ℓ} {X}) Embedding→Subset→Embedding {ℓ = ℓ} {X = X} (A , f , ψ) = - cong (equivFun Σ-assoc-≃) (Σ≡Prop (λ _ → isPropIsEmbedding) (retEq (fibrationEquiv X ℓ) (A , f))) + cong (equivFun Σ-assoc-≃) (Σ≡Prop (λ _ → isPropIsEmbedding) (retEq (fibrationEquiv X ℓ) (A , f))) Subset≃Embedding : {X : Type ℓ} → ℙ X ≃ (Σ[ A ∈ Type ℓ ] (A ↪ X)) Subset≃Embedding = isoToEquiv (iso Subset→Embedding Embedding→Subset @@ -282,7 +282,7 @@ Cod′ : ∀ u v → Type _ Cod′ u v = Σ[ p ∈ f (fst u) ≡ f (fst v) ] PathP (λ i → p i ≡ z) (snd u) (snd v) ΣeqCf : Dom′ u v ≃ Cod′ u v - ΣeqCf = Σ-cong-equiv-fst (_ , isE _ _) + ΣeqCf = Σ-cong-equiv-fst (_ , isE _ _) dom→ : u ≡ v → Dom′ u v dom→ p = cong fst p , cong snd p @@ -321,7 +321,7 @@ (B ≃ A) ≃⟨ invEquiv univalence ⟩ (B ≡ A) ■ fiberSingl : ∀ X → fiber F (F X) ≃ singl X - fiberSingl X = Σ-cong-equiv-snd (λ _ → lemma _ _) + fiberSingl X = Σ-cong-equiv-snd (λ _ → lemma _ _) propFibersF : hasPropFibersOfImage F propFibersF X = Embedding-into-isProp→isProp (Equiv→Embedding (fiberSingl X)) isPropSingl @@ -368,11 +368,11 @@ ≃ (Σ[ (E , eq) ∈ fiber L A ] fiber (_∘ lower) (transport⁻ (λ i → eq i → B) f)) fiberChar = fiber liftFibration (A , f) - ≃⟨ Σ-cong-equiv-snd (λ _ → invEquiv ΣPath≃PathΣ) ⟩ + ≃⟨ Σ-cong-equiv-snd (λ _ → invEquiv ΣPath≃PathΣ) ⟩ (Σ[ (E , g) ∈ Fibration B ℓ' ] Σ[ eq ∈ (L E ≡ A) ] PathP (λ i → eq i → B) (g ∘ lower) f) ≃⟨ boringSwap ⟩ (Σ[ (E , eq) ∈ fiber L A ] Σ[ g ∈ (E → B) ] PathP (λ i → eq i → B) (g ∘ lower) f) - ≃⟨ Σ-cong-equiv-snd (λ _ → Σ-cong-equiv-snd λ _ → pathToEquiv (PathP≡Path⁻ _ _ _)) ⟩ + ≃⟨ Σ-cong-equiv-snd (λ _ → Σ-cong-equiv-snd λ _ → pathToEquiv (PathP≡Path⁻ _ _ _)) ⟩ (Σ[ (E , eq) ∈ fiber L A ] fiber (_∘ lower) (transport⁻ (λ i → eq i → B) f)) ■ where unquoteDecl boringSwap = @@ -391,8 +391,8 @@ FibrationIP : f≃g ≃ (f ≡ g) FibrationIP = - f≃g ≃⟨ equivΠCod (λ b → equivComp (Σ-cong-equiv-fst LiftEquiv) - (Σ-cong-equiv-fst LiftEquiv)) ⟩ + f≃g ≃⟨ equivΠCod (λ b → equivComp (Σ-cong-equiv-fst LiftEquiv) + (Σ-cong-equiv-fst LiftEquiv)) ⟩ f≃g′ ≃⟨ Fibration′IP ⟩ (liftFibration f ≡ liftFibration g) ≃⟨ invEquiv (_ , isEmbeddingLiftFibration _ _) ⟩ (f ≡ g) ■ @@ -421,12 +421,12 @@ isEmbeddingToFibr w x = fullEquiv .snd where -- carefully managed such that (cong toFibr) is the equivalence fullEquiv : (w ≡ x) ≃ (toFibr w ≡ toFibr x) - fullEquiv = compEquiv (congEquiv (invEquiv Σ-assoc-≃)) (invEquiv (Σ≡PropEquiv (λ _ → isPropIsEmbedding))) + fullEquiv = compEquiv (congEquiv (invEquiv Σ-assoc-≃)) (invEquiv (Σ≡PropEquiv (λ _ → isPropIsEmbedding))) EmbeddingIP : f≃g ≃ (f ≡ g) EmbeddingIP = f≃g - ≃⟨ strictIsoToEquiv (invIso toProdIso) ⟩ + ≃⟨ strictIsoToEquiv (invIso toProdIso) ⟩ (∀ b → (fiber ffun b → fiber gfun b) × (fiber gfun b → fiber ffun b)) ≃⟨ equivΠCod (λ _ → isEquivPropBiimpl→Equiv (isEmbedding→hasPropFibers isEmbF _) (isEmbedding→hasPropFibers isEmbG _)) ⟩ @@ -466,7 +466,7 @@ equiv : ∀ x y → isEquiv (apmap x y) equiv x y = ((invEquiv ΣPathP≃PathPΣ) - ∙ₑ (≃-× ((cong f) , (embf (fst x) (fst y))) + ∙ₑ (≃-× ((cong f) , (embf (fst x) (fst y))) ((cong g) , (embg (snd x) (snd y)))) ∙ₑ ΣPathP≃PathPΣ) .snd @@ -474,5 +474,5 @@ emb x y = equiv x y EmbeddingΣProp : {A : Type ℓ} → {B : A → Type ℓ'} → (∀ a → isProp (B a)) → Σ A B ↪ A -EmbeddingΣProp f = fst , (λ _ _ → isEmbeddingFstΣProp f) +EmbeddingΣProp f = fst , (λ _ _ → isEmbeddingFstΣProp f) \ No newline at end of file diff --git a/Cubical.Functions.Image.html b/Cubical.Functions.Image.html index 76bf67838f..fc3b691e00 100644 --- a/Cubical.Functions.Image.html +++ b/Cubical.Functions.Image.html @@ -36,7 +36,7 @@ λ y → isOfHLevelRetractFromIso 1 (ϕ y) isPropPropTrunc where ϕ : (y : B) → Iso _ _ - ϕ y = invIso (fiberProjIso B isInImage y) + ϕ y = invIso (fiberProjIso B isInImage y) restrictToImage : A → Image restrictToImage x = (f x) , ∣ x , refl ∣₁ @@ -45,7 +45,7 @@ isSurjectionImageRestriction (y , y∈im) = PT.rec isPropPropTrunc (λ (x , fx≡y) - → ∣ x , Σ≡Prop isPropIsInImage fx≡y ∣₁) + → ∣ x , Σ≡Prop isPropIsInImage fx≡y ∣₁) y∈im imageFactorization : fst imageInclusion ∘ restrictToImage ≡ f diff --git a/Cubical.HITs.Cost.Base.html b/Cubical.HITs.Cost.Base.html index b08b800b03..a4be4c9bfa 100644 --- a/Cubical.HITs.Cost.Base.html +++ b/Cubical.HITs.Cost.Base.html @@ -18,7 +18,7 @@ -- To compare two elements of Cost A we only need to look at the first parts Cost≡ : (x y : Cost A) → x .fst ≡ y .fst → x ≡ y -Cost≡ _ _ = Σ≡Prop λ _ → squash₁ +Cost≡ _ _ = Σ≡Prop λ _ → squash₁ -- To make it easier to program with Cost A we prove that it forms a -- monad which counts the number of calls to >>=. We could also turn diff --git a/Cubical.HITs.CumulativeHierarchy.Properties.html b/Cubical.HITs.CumulativeHierarchy.Properties.html index 47d7cf4a71..0534453e33 100644 --- a/Cubical.HITs.CumulativeHierarchy.Properties.html +++ b/Cubical.HITs.CumulativeHierarchy.Properties.html @@ -123,7 +123,7 @@ repFiber f b = Σ[ a ∈ _ ] f a ≊ b repFiber≃fiber : (f : X → V ℓ) (b : V ℓ) → repFiber f b ≃ fiber f b -repFiber≃fiber f b = Σ-cong-equiv (idEquiv _) (λ _ → identityPrinciple) +repFiber≃fiber f b = Σ-cong-equiv (idEquiv _) (λ _ → identityPrinciple) -- projecting out a representing type together with the embedding MonicPresentation : (a : V ℓ) → Type (ℓ-suc ℓ) @@ -198,9 +198,9 @@ V-fixpoint : V ℓ ≃ MonicDataF (V ℓ) V-fixpoint {ℓ} = - V ℓ ≃⟨ invEquiv (Σ-contractSnd λ a → inhProp→isContr (V-repr a) (isPropMonicPresentation a)) ⟩ + V ℓ ≃⟨ invEquiv (Σ-contractSnd λ a → inhProp→isContr (V-repr a) (isPropMonicPresentation a)) ⟩ (Σ[ a ∈ V ℓ ] MonicPresentation a) ≃⟨ boringswap ⟩ - (Σ[ (X , ix , _) ∈ MonicDataF (V ℓ) ] singl (sett X ix)) ≃⟨ Σ-contractSnd (λ _ → isContrSingl _) ⟩ + (Σ[ (X , ix , _) ∈ MonicDataF (V ℓ) ] singl (sett X ix)) ≃⟨ Σ-contractSnd (λ _ → isContrSingl _) ⟩ MonicDataF (V ℓ) ■ where boringswap : (Σ[ a ∈ V ℓ ] MonicPresentation a) ≃ (Σ[ (X , ix , _) ∈ MonicDataF (V ℓ) ] singl (sett X ix)) boringswap = isoToEquiv (iso @@ -274,7 +274,7 @@ from : ⟪ a ⟫ → Σ[ v ∈ V _ ] ⟨ v ∈ₛ a ⟩ from ⟪a⟫ = ⟪ a ⟫↪ ⟪a⟫ , ∈ₛ⟪ a ⟫↪ ⟪a⟫ retr : retract into from - retr s = Σ≡Prop (λ v → (v ∈ₛ a) .snd) (equivFun identityPrinciple (s .snd .snd)) + retr s = Σ≡Prop (λ v → (v ∈ₛ a) .snd) (equivFun identityPrinciple (s .snd .snd)) -- subset relation, once in level (ℓ-suc ℓ) and once in ℓ _⊆_ : (a b : V ℓ) → hProp (ℓ-suc ℓ) @@ -288,7 +288,7 @@ ⊆⇔⊆ₛ : (a b : V ℓ) → ⟨ a ⊆ b ⇔ a ⊆ₛ b ⟩ ⊆⇔⊆ₛ a b = - (λ s → invEq curryEquiv s ∘ invEq (presentation a)) + (λ s → invEq curryEquiv s ∘ invEq (presentation a)) , (λ s x xa → subst (λ x → ⟨ x ∈ₛ b ⟩) (equivFun identityPrinciple (xa .snd)) (s (xa .fst))) -- the homotopy definition of equality as an hProp, we know this is equivalent to bisimulation diff --git a/Cubical.HITs.FreeGroup.Properties.html b/Cubical.HITs.FreeGroup.Properties.html index 1102262be6..a353eb8a81 100644 --- a/Cubical.HITs.FreeGroup.Properties.html +++ b/Cubical.HITs.FreeGroup.Properties.html @@ -22,182 +22,211 @@ open import Cubical.Foundations.Function open import Cubical.Foundations.Equiv open import Cubical.Foundations.Equiv.BiInvertible - -open import Cubical.Algebra.Group -open import Cubical.Algebra.Group.Morphisms -open import Cubical.Algebra.Group.MorphismProperties -open import Cubical.Algebra.Monoid.Base -open import Cubical.Algebra.Semigroup.Base - -private - variable - ℓ ℓ' : Level - A : Type ℓ - -freeGroupIsSet : isSet (FreeGroup A) -freeGroupIsSet = trunc - -freeGroupIsSemiGroup : IsSemigroup {A = FreeGroup A} _·_ -freeGroupIsSemiGroup = issemigroup freeGroupIsSet assoc - -freeGroupIsMonoid : IsMonoid {A = FreeGroup A} ε _·_ -freeGroupIsMonoid = ismonoid freeGroupIsSemiGroup (λ x → sym (idr x)) (λ x → sym (idl x)) - -freeGroupIsGroup : IsGroup {G = FreeGroup A} ε _·_ inv -freeGroupIsGroup = isgroup freeGroupIsMonoid invr invl - -freeGroupGroupStr : GroupStr (FreeGroup A) -freeGroupGroupStr = groupstr ε _·_ inv freeGroupIsGroup - --- FreeGroup is indeed a group -freeGroupGroup : Type ℓ → Group ℓ -freeGroupGroup A = FreeGroup A , freeGroupGroupStr - --- The recursion principle for the FreeGroup -rec : {Group : Group ℓ'} → (A → Group .fst) → GroupHom (freeGroupGroup A) Group -rec {Group = G , groupstr 1g _·g_ invg (isgroup (ismonoid isSemigroupg ·gIdR ·gIdL) ·gInvR ·gInvL)} f = f' , isHom - where - f' : FreeGroup _ → G - f' (η a) = f a - f' (g1 · g2) = (f' g1) ·g (f' g2) - f' ε = 1g - f' (inv g) = invg (f' g) - f' (assoc g1 g2 g3 i) = IsSemigroup.·Assoc isSemigroupg (f' g1) (f' g2) (f' g3) i - f' (idr g i) = sym (·gIdR (f' g)) i - f' (idl g i) = sym (·gIdL (f' g)) i - f' (invr g i) = ·gInvR (f' g) i - f' (invl g i) = ·gInvL (f' g) i - f' (trunc g1 g2 p q i i₁) = IsSemigroup.is-set isSemigroupg (f' g1) (f' g2) (cong f' p) (cong f' q) i i₁ - - isHom : IsGroupHom freeGroupGroupStr f' _ - IsGroupHom.pres· isHom = λ g1 g2 → refl - IsGroupHom.pres1 isHom = refl - IsGroupHom.presinv isHom = λ g → refl - --- The induction principle for the FreeGroup for hProps -elimProp : {B : FreeGroup A → Type ℓ'} - → ((x : FreeGroup A) → isProp (B x)) - → ((a : A) → B (η a)) - → ((g1 g2 : FreeGroup A) → B g1 → B g2 → B (g1 · g2)) - → (B ε) - → ((g : FreeGroup A) → B g → B (inv g)) - → (x : FreeGroup A) - → B x -elimProp {B = B} Bprop η-ind ·-ind ε-ind inv-ind = induction where - induction : ∀ g → B g - induction (η a) = η-ind a - induction (g1 · g2) = ·-ind g1 g2 (induction g1) (induction g2) - induction ε = ε-ind - induction (inv g) = inv-ind g (induction g) - induction (assoc g1 g2 g3 i) = path i where - p1 : B g1 - p1 = induction g1 - p2 : B g2 - p2 = induction g2 - p3 : B g3 - p3 = induction g3 - path : PathP (λ i → B (assoc g1 g2 g3 i)) (·-ind g1 (g2 · g3) p1 (·-ind g2 g3 p2 p3)) - (·-ind (g1 · g2) g3 (·-ind g1 g2 p1 p2) p3) - path = isProp→PathP (λ i → Bprop (assoc g1 g2 g3 i)) (·-ind g1 (g2 · g3) p1 (·-ind g2 g3 p2 p3)) - (·-ind (g1 · g2) g3 (·-ind g1 g2 p1 p2) p3) - induction (idr g i) = path i where - p : B g - p = induction g - pε : B ε - pε = induction ε - path : PathP (λ i → B (idr g i)) p (·-ind g ε p pε) - path = isProp→PathP (λ i → Bprop (idr g i)) p (·-ind g ε p pε) - induction (idl g i) = path i where - p : B g - p = induction g - pε : B ε - pε = induction ε - path : PathP (λ i → B (idl g i)) p (·-ind ε g pε p) - path = isProp→PathP (λ i → Bprop (idl g i)) p (·-ind ε g pε p) - induction (invr g i) = path i where - p : B g - p = induction g - pinv : B (inv g) - pinv = inv-ind g p - pε : B ε - pε = induction ε - path : PathP (λ i → B (invr g i)) (·-ind g (inv g) p pinv) pε - path = isProp→PathP (λ i → Bprop (invr g i)) (·-ind g (inv g) p pinv) pε - induction (invl g i) = path i where - p : B g - p = induction g - pinv : B (inv g) - pinv = inv-ind g p - pε : B ε - pε = induction ε - path : PathP (λ i → B (invl g i)) (·-ind (inv g) g pinv p) pε - path = isProp→PathP (λ i → Bprop (invl g i)) (·-ind (inv g) g pinv p) pε - induction (trunc g1 g2 q1 q2 i i₁) = square i i₁ where - p1 : B g1 - p1 = induction g1 - p2 : B g2 - p2 = induction g2 - dq1 : PathP (λ i → B (q1 i)) p1 p2 - dq1 = cong induction q1 - dq2 : PathP (λ i → B (q2 i)) p1 p2 - dq2 = cong induction q2 - square : SquareP (λ i i₁ → B (trunc g1 g2 q1 q2 i i₁)) - {a₀₀ = p1} {a₀₁ = p2} dq1 - {a₁₀ = p1} {a₁₁ = p2} dq2 - (cong induction (refl {x = g1})) (cong induction (refl {x = g2})) - square = isProp→SquareP (λ i i₁ → Bprop (trunc g1 g2 q1 q2 i i₁)) - (cong induction (refl {x = g1})) (cong induction (refl {x = g2})) - dq1 dq2 - --- Two group homomorphisms from FreeGroup to G are the same if they agree on every a : A -freeGroupHom≡ : {Group : Group ℓ'}{f g : GroupHom (freeGroupGroup A) Group} - → (∀ a → (fst f) (η a) ≡ (fst g) (η a)) → f ≡ g -freeGroupHom≡ {Group = G , (groupstr 1g _·g_ invg isGrp)} {f} {g} eqOnA = GroupHom≡ (funExt pointwise) where - open IsGroup isGrp using (is-set) - B : ∀ x → Type _ - B x = (fst f) x ≡ (fst g) x - Bprop : ∀ x → isProp (B x) - Bprop x = is-set ((fst f) x) ((fst g) x) - η-ind : ∀ a → B (η a) - η-ind = eqOnA - ·-ind : ∀ g1 g2 → B g1 → B g2 → B (g1 · g2) - ·-ind g1 g2 ind1 ind2 = - (fst f) (g1 · g2) - ≡⟨ IsGroupHom.pres· (f .snd) g1 g2 ⟩ - ((fst f) g1) ·g ((fst f) g2) - ≡⟨ cong (λ x → x ·g ((fst f) g2)) ind1 ⟩ - ((fst g) g1) ·g ((fst f) g2) - ≡⟨ cong (λ x → ((fst g) g1) ·g x) ind2 ⟩ - ((fst g) g1) ·g ((fst g) g2) - ≡⟨ sym (IsGroupHom.pres· (g .snd) g1 g2) ⟩ - (fst g) (g1 · g2) ∎ - ε-ind : B ε - ε-ind = - (fst f) ε - ≡⟨ IsGroupHom.pres1 (f .snd) ⟩ - 1g - ≡⟨ sym (IsGroupHom.pres1 (g .snd)) ⟩ - (fst g) ε ∎ - inv-ind : ∀ x → B x → B (inv x) - inv-ind x ind = - (fst f) (inv x) - ≡⟨ IsGroupHom.presinv (f .snd) x ⟩ - invg ((fst f) x) - ≡⟨ cong invg ind ⟩ - invg ((fst g) x) - ≡⟨ sym (IsGroupHom.presinv (g .snd) x) ⟩ - (fst g) (inv x) ∎ - pointwise : ∀ x → (fst f) x ≡ (fst g) x - pointwise = elimProp Bprop η-ind ·-ind ε-ind inv-ind - --- The type of Group Homomorphisms from the FreeGroup A into G --- is equivalent to the type of functions from A into G .fst -A→Group≃GroupHom : {Group : Group ℓ'} → (A → Group .fst) ≃ GroupHom (freeGroupGroup A) Group -A→Group≃GroupHom {Group = Group} = biInvEquiv→Equiv-right biInv where - biInv : BiInvEquiv (A → Group .fst) (GroupHom (freeGroupGroup A) Group) - BiInvEquiv.fun biInv = rec - BiInvEquiv.invr biInv (hom , _) a = hom (η a) - BiInvEquiv.invr-rightInv biInv hom = freeGroupHom≡ (λ a → refl) - BiInvEquiv.invl biInv (hom , _) a = hom (η a) - BiInvEquiv.invl-leftInv biInv f = funExt (λ a → refl) +open import Cubical.Foundations.Isomorphism + +open import Cubical.Algebra.Group +open import Cubical.Algebra.Group.Morphisms +open import Cubical.Algebra.Group.MorphismProperties +open import Cubical.Algebra.Monoid.Base +open import Cubical.Algebra.Semigroup.Base + +open import Cubical.Categories.Functor.Base +open import Cubical.Categories.Adjoint +open import Cubical.Categories.Instances.Sets +open import Cubical.Categories.Instances.Groups + +import Cubical.Data.Int as ℤ +open import Cubical.Algebra.Group.Instances.Int + +private + variable + ℓ ℓ' : Level + A : Type ℓ + +freeGroupIsSet : isSet (FreeGroup A) +freeGroupIsSet = trunc + +freeGroupIsSemiGroup : IsSemigroup {A = FreeGroup A} _·_ +freeGroupIsSemiGroup = issemigroup freeGroupIsSet assoc + +freeGroupIsMonoid : IsMonoid {A = FreeGroup A} ε _·_ +freeGroupIsMonoid = ismonoid freeGroupIsSemiGroup (λ x → sym (idr x)) (λ x → sym (idl x)) + +freeGroupIsGroup : IsGroup {G = FreeGroup A} ε _·_ inv +freeGroupIsGroup = isgroup freeGroupIsMonoid invr invl + +freeGroupGroupStr : GroupStr (FreeGroup A) +freeGroupGroupStr = groupstr ε _·_ inv freeGroupIsGroup + +-- FreeGroup is indeed a group +freeGroupGroup : Type ℓ → Group ℓ +freeGroupGroup A = FreeGroup A , freeGroupGroupStr + +-- The recursion principle for the FreeGroup +rec : {Group : Group ℓ'} → (A → Group .fst) → GroupHom (freeGroupGroup A) Group +rec {Group = G , groupstr 1g _·g_ invg (isgroup (ismonoid isSemigroupg ·gIdR ·gIdL) ·gInvR ·gInvL)} f = f' , isHom + where + f' : FreeGroup _ → G + f' (η a) = f a + f' (g1 · g2) = (f' g1) ·g (f' g2) + f' ε = 1g + f' (inv g) = invg (f' g) + f' (assoc g1 g2 g3 i) = IsSemigroup.·Assoc isSemigroupg (f' g1) (f' g2) (f' g3) i + f' (idr g i) = sym (·gIdR (f' g)) i + f' (idl g i) = sym (·gIdL (f' g)) i + f' (invr g i) = ·gInvR (f' g) i + f' (invl g i) = ·gInvL (f' g) i + f' (trunc g1 g2 p q i i₁) = IsSemigroup.is-set isSemigroupg (f' g1) (f' g2) (cong f' p) (cong f' q) i i₁ + + isHom : IsGroupHom freeGroupGroupStr f' _ + IsGroupHom.pres· isHom = λ g1 g2 → refl + IsGroupHom.pres1 isHom = refl + IsGroupHom.presinv isHom = λ g → refl + +-- The induction principle for the FreeGroup for hProps +elimProp : {B : FreeGroup A → Type ℓ'} + → ((x : FreeGroup A) → isProp (B x)) + → ((a : A) → B (η a)) + → ((g1 g2 : FreeGroup A) → B g1 → B g2 → B (g1 · g2)) + → (B ε) + → ((g : FreeGroup A) → B g → B (inv g)) + → (x : FreeGroup A) + → B x +elimProp {B = B} Bprop η-ind ·-ind ε-ind inv-ind = induction where + induction : ∀ g → B g + induction (η a) = η-ind a + induction (g1 · g2) = ·-ind g1 g2 (induction g1) (induction g2) + induction ε = ε-ind + induction (inv g) = inv-ind g (induction g) + induction (assoc g1 g2 g3 i) = path i where + p1 : B g1 + p1 = induction g1 + p2 : B g2 + p2 = induction g2 + p3 : B g3 + p3 = induction g3 + path : PathP (λ i → B (assoc g1 g2 g3 i)) (·-ind g1 (g2 · g3) p1 (·-ind g2 g3 p2 p3)) + (·-ind (g1 · g2) g3 (·-ind g1 g2 p1 p2) p3) + path = isProp→PathP (λ i → Bprop (assoc g1 g2 g3 i)) (·-ind g1 (g2 · g3) p1 (·-ind g2 g3 p2 p3)) + (·-ind (g1 · g2) g3 (·-ind g1 g2 p1 p2) p3) + induction (idr g i) = path i where + p : B g + p = induction g + pε : B ε + pε = induction ε + path : PathP (λ i → B (idr g i)) p (·-ind g ε p pε) + path = isProp→PathP (λ i → Bprop (idr g i)) p (·-ind g ε p pε) + induction (idl g i) = path i where + p : B g + p = induction g + pε : B ε + pε = induction ε + path : PathP (λ i → B (idl g i)) p (·-ind ε g pε p) + path = isProp→PathP (λ i → Bprop (idl g i)) p (·-ind ε g pε p) + induction (invr g i) = path i where + p : B g + p = induction g + pinv : B (inv g) + pinv = inv-ind g p + pε : B ε + pε = induction ε + path : PathP (λ i → B (invr g i)) (·-ind g (inv g) p pinv) pε + path = isProp→PathP (λ i → Bprop (invr g i)) (·-ind g (inv g) p pinv) pε + induction (invl g i) = path i where + p : B g + p = induction g + pinv : B (inv g) + pinv = inv-ind g p + pε : B ε + pε = induction ε + path : PathP (λ i → B (invl g i)) (·-ind (inv g) g pinv p) pε + path = isProp→PathP (λ i → Bprop (invl g i)) (·-ind (inv g) g pinv p) pε + induction (trunc g1 g2 q1 q2 i i₁) = square i i₁ where + p1 : B g1 + p1 = induction g1 + p2 : B g2 + p2 = induction g2 + dq1 : PathP (λ i → B (q1 i)) p1 p2 + dq1 = cong induction q1 + dq2 : PathP (λ i → B (q2 i)) p1 p2 + dq2 = cong induction q2 + square : SquareP (λ i i₁ → B (trunc g1 g2 q1 q2 i i₁)) + {a₀₀ = p1} {a₀₁ = p2} dq1 + {a₁₀ = p1} {a₁₁ = p2} dq2 + (cong induction (refl {x = g1})) (cong induction (refl {x = g2})) + square = isProp→SquareP (λ i i₁ → Bprop (trunc g1 g2 q1 q2 i i₁)) + (cong induction (refl {x = g1})) (cong induction (refl {x = g2})) + dq1 dq2 + +-- Two group homomorphisms from FreeGroup to G are the same if they agree on every a : A +freeGroupHom≡ : {Group : Group ℓ'}{f g : GroupHom (freeGroupGroup A) Group} + → (∀ a → (fst f) (η a) ≡ (fst g) (η a)) → f ≡ g +freeGroupHom≡ {Group = G , (groupstr 1g _·g_ invg isGrp)} {f} {g} eqOnA = GroupHom≡ (funExt pointwise) where + open IsGroup isGrp using (is-set) + B : ∀ x → Type _ + B x = (fst f) x ≡ (fst g) x + Bprop : ∀ x → isProp (B x) + Bprop x = is-set ((fst f) x) ((fst g) x) + η-ind : ∀ a → B (η a) + η-ind = eqOnA + ·-ind : ∀ g1 g2 → B g1 → B g2 → B (g1 · g2) + ·-ind g1 g2 ind1 ind2 = + (fst f) (g1 · g2) + ≡⟨ IsGroupHom.pres· (f .snd) g1 g2 ⟩ + ((fst f) g1) ·g ((fst f) g2) + ≡⟨ cong (λ x → x ·g ((fst f) g2)) ind1 ⟩ + ((fst g) g1) ·g ((fst f) g2) + ≡⟨ cong (λ x → ((fst g) g1) ·g x) ind2 ⟩ + ((fst g) g1) ·g ((fst g) g2) + ≡⟨ sym (IsGroupHom.pres· (g .snd) g1 g2) ⟩ + (fst g) (g1 · g2) ∎ + ε-ind : B ε + ε-ind = + (fst f) ε + ≡⟨ IsGroupHom.pres1 (f .snd) ⟩ + 1g + ≡⟨ sym (IsGroupHom.pres1 (g .snd)) ⟩ + (fst g) ε ∎ + inv-ind : ∀ x → B x → B (inv x) + inv-ind x ind = + (fst f) (inv x) + ≡⟨ IsGroupHom.presinv (f .snd) x ⟩ + invg ((fst f) x) + ≡⟨ cong invg ind ⟩ + invg ((fst g) x) + ≡⟨ sym (IsGroupHom.presinv (g .snd) x) ⟩ + (fst g) (inv x) ∎ + pointwise : ∀ x → (fst f) x ≡ (fst g) x + pointwise = elimProp Bprop η-ind ·-ind ε-ind inv-ind + +-- The type of Group Homomorphisms from the FreeGroup A into G +-- is equivalent to the type of functions from A into G .fst +A→Group≃GroupHom : {Group : Group ℓ'} → (A → Group .fst) ≃ GroupHom (freeGroupGroup A) Group +A→Group≃GroupHom {Group = Group} = biInvEquiv→Equiv-right biInv where + biInv : BiInvEquiv (A → Group .fst) (GroupHom (freeGroupGroup A) Group) + BiInvEquiv.fun biInv = rec + BiInvEquiv.invr biInv (hom , _) a = hom (η a) + BiInvEquiv.invr-rightInv biInv hom = freeGroupHom≡ (λ a → refl) + BiInvEquiv.invl biInv (hom , _) a = hom (η a) + BiInvEquiv.invl-leftInv biInv f = funExt (λ a → refl) + +freeGroupFunctor : Functor (SET ℓ) (GroupCategory {ℓ}) +Functor.F-ob freeGroupFunctor (A , _) = freeGroupGroup A +Functor.F-hom freeGroupFunctor f = rec (η ∘ f) +Functor.F-id freeGroupFunctor = + freeGroupHom≡ λ _ → refl +Functor.F-seq freeGroupFunctor _ _ = + freeGroupHom≡ λ _ → refl + +freeGroupFunctor⊣Forget : freeGroupFunctor NaturalBijection.⊣ (Forget {ℓ}) +NaturalBijection._⊣_.adjIso freeGroupFunctor⊣Forget = + invIso (equivToIso A→Group≃GroupHom) +NaturalBijection._⊣_.adjNatInD freeGroupFunctor⊣Forget _ _ = refl +NaturalBijection._⊣_.adjNatInC freeGroupFunctor⊣Forget _ _ = freeGroupHom≡ λ _ → refl + +windingHom : GroupHom (freeGroupGroup A) ℤGroup +windingHom = rec λ _ → ℤ.pos 1 + +winding : FreeGroup A → ℤ.ℤ +winding = fst windingHom \ No newline at end of file diff --git a/Cubical.HITs.FreeGroupoid.Properties.html b/Cubical.HITs.FreeGroupoid.Properties.html index 31bff6578f..72e12c1578 100644 --- a/Cubical.HITs.FreeGroupoid.Properties.html +++ b/Cubical.HITs.FreeGroupoid.Properties.html @@ -14,7 +14,7 @@ module Cubical.HITs.FreeGroupoid.Properties where open import Cubical.HITs.FreeGroupoid.Base -open import Cubical.HITs.FreeGroup renaming (elimProp to freeGroupElimProp) +open import Cubical.HITs.FreeGroup renaming (elimProp to freeGroupElimProp) open import Cubical.Foundations.Prelude open import Cubical.Foundations.Equiv @@ -132,13 +132,13 @@ ∥freeGroupoid∥₂Group A = ∥ FreeGroupoid A ∥₂ , ∥freeGroupoid∥₂GroupStr -forgetfulHom : GroupHom (freeGroupGroup A) (∥freeGroupoid∥₂Group A) -forgetfulHom = rec (λ a → ∣ η a ∣₂) +forgetfulHom : GroupHom (freeGroupGroup A) (∥freeGroupoid∥₂Group A) +forgetfulHom = rec (λ a → ∣ η a ∣₂) -forgetfulHom⁻¹ : GroupHom (∥freeGroupoid∥₂Group A) (freeGroupGroup A) +forgetfulHom⁻¹ : GroupHom (∥freeGroupoid∥₂Group A) (freeGroupGroup A) forgetfulHom⁻¹ = invf , isHom where invf : ∥ FreeGroupoid A ∥₂ → FreeGroup A - invf = recTrunc freeGroupIsSet aux where + invf = recTrunc freeGroupIsSet aux where aux : FreeGroupoid A → FreeGroup A aux (η a) = η a aux (g1 · g2) = (aux g1) · (aux g2) @@ -149,12 +149,12 @@ aux (idl g i) = idl (aux g) i aux (invr g i) = invr (aux g) i aux (invl g i) = invl (aux g) i - isHom : IsGroupHom {A = ∥ FreeGroupoid A ∥₂} {B = FreeGroup A} ∥freeGroupoid∥₂GroupStr invf freeGroupGroupStr - IsGroupHom.pres· isHom x y = elim2 (λ x y → isProp→isSet (freeGroupIsSet (invf (x ∣·∣₂ y)) ((invf x) · (invf y)))) ind x y where + isHom : IsGroupHom {A = ∥ FreeGroupoid A ∥₂} {B = FreeGroup A} ∥freeGroupoid∥₂GroupStr invf freeGroupGroupStr + IsGroupHom.pres· isHom x y = elim2 (λ x y → isProp→isSet (freeGroupIsSet (invf (x ∣·∣₂ y)) ((invf x) · (invf y)))) ind x y where ind : ∀ g1 g2 → invf (∣ g1 ∣₂ ∣·∣₂ ∣ g2 ∣₂) ≡ (invf ∣ g1 ∣₂) · (invf ∣ g2 ∣₂) ind g1 g2 = refl IsGroupHom.pres1 isHom = refl - IsGroupHom.presinv isHom x = elim (λ x → isProp→isSet (freeGroupIsSet (invf (∣inv∣₂ x)) (inv (invf x)))) ind x where + IsGroupHom.presinv isHom x = elim (λ x → isProp→isSet (freeGroupIsSet (invf (∣inv∣₂ x)) (inv (invf x)))) ind x where ind : ∀ g → invf (∣inv∣₂ ∣ g ∣₂) ≡ inv (invf ∣ g ∣₂) ind g = refl @@ -202,7 +202,7 @@ leftInv : ∀ (g : FreeGroup A) → invf (f g) ≡ g leftInv g = freeGroupElimProp Bprop η-ind ·-ind ε-ind inv-ind g where Bprop : ∀ g → isProp (invf (f g) ≡ g) - Bprop g = freeGroupIsSet (invf (f g)) g + Bprop g = freeGroupIsSet (invf (f g)) g η-ind : ∀ a → invf (f (η a)) ≡ (η a) η-ind a = refl ·-ind : ∀ g1 g2 → invf (f g1) ≡ g1 → invf (f g2) ≡ g2 → invf (f (g1 · g2)) ≡ g1 · g2 diff --git a/Cubical.HITs.James.Inductive.PushoutFormula.html b/Cubical.HITs.James.Inductive.PushoutFormula.html index 1a3c2353b4..a45345f0e5 100644 --- a/Cubical.HITs.James.Inductive.PushoutFormula.html +++ b/Cubical.HITs.James.Inductive.PushoutFormula.html @@ -155,7 +155,7 @@ private left≃ : X × 𝕁ames 1 ≃ X × X - left≃ = ≃-× (idEquiv _) 𝕁ames1≃ + left≃ = ≃-× (idEquiv _) 𝕁ames1≃ lComm : (x : 𝕁amesPush 0) → left≃ .fst (leftMap x) ≡ ⋁↪ (P0→X⋁X x) lComm (inl (x , [])) = refl diff --git a/Cubical.HITs.James.LoopSuspEquiv.html b/Cubical.HITs.James.LoopSuspEquiv.html index 39d4723222..6c5660b87d 100644 --- a/Cubical.HITs.James.LoopSuspEquiv.html +++ b/Cubical.HITs.James.LoopSuspEquiv.html @@ -117,7 +117,7 @@ open FlatteningLemma (λ _ → tt) (λ _ → tt) (λ tt → James) (λ tt → James) (λ x → _ , isEquiv∷ x) Total≃ : Pushout Σf Σg ≃ Total - Total≃ = pushoutEquiv _ _ _ _ (idEquiv _) (ΣUnit _) (ΣUnit _) refl refl + Total≃ = pushoutEquiv _ _ _ _ (idEquiv _) (ΣUnit _) (ΣUnit _) refl refl PushoutSuspCode : (x : PushoutSusp X) → E x ≃ Code (PushoutSusp→Susp x) PushoutSuspCode (inl tt) = idEquiv _ @@ -125,7 +125,7 @@ PushoutSuspCode (push x i) = idEquiv _ ΣCode≃' : _ ≃ Σ _ Code - ΣCode≃' = Σ-cong-equiv PushoutSusp≃Susp PushoutSuspCode + ΣCode≃' = Σ-cong-equiv PushoutSusp≃Susp PushoutSuspCode ΣCode≃ : Total ≃ Σ _ Code ΣCode≃ = compEquiv (invEquiv Total≃) (compEquiv (invEquiv flatten) ΣCode≃') diff --git a/Cubical.HITs.PropositionalTruncation.Properties.html b/Cubical.HITs.PropositionalTruncation.Properties.html index 1c1d301103..f10c3bf48b 100644 --- a/Cubical.HITs.PropositionalTruncation.Properties.html +++ b/Cubical.HITs.PropositionalTruncation.Properties.html @@ -216,7 +216,7 @@ eqv : (g : Σ (A → B) 2-Constant) → ∀ fi → fib g ≡ fi eqv g (f , p) = - Σ≡Prop (λ f → isOfHLevelΣ 2 Fset Kset _ _) + Σ≡Prop (λ f → isOfHLevelΣ 2 Fset Kset _ _) (cong (uncurry rec→Set) (sym p) ∙ setRecLemma f) trunc→Set≃ : (∥ A ∥₁ → B) ≃ (Σ (A → B) 2-Constant) diff --git a/Cubical.HITs.Pushout.KrausVonRaumer.html b/Cubical.HITs.Pushout.KrausVonRaumer.html index 659bdbbf66..8777d63538 100644 --- a/Cubical.HITs.Pushout.KrausVonRaumer.html +++ b/Cubical.HITs.Pushout.KrausVonRaumer.html @@ -149,7 +149,7 @@ subst (P (f a) p ≃_) (cong (λ w → fiber (cong ⊔.inr) (p ∙ w)) (lUnit (push a) ⁻¹)) - (Σ-contractFst (inhProp→isContr (a , refl) (isEmbedding→hasPropFibers fEmb (f a))))) + (Σ-contractFst (inhProp→isContr (a , refl) (isEmbedding→hasPropFibers fEmb (f a))))) bwd : ∀ c → (t : ⊔.inr c₀ ≡ ⊔.inr c) → fiber (cong ⊔.inr) t bwd = Bwd.elimR diff --git a/Cubical.HITs.RPn.Base.html b/Cubical.HITs.RPn.Base.html index 0a2d4d89cf..1ec9b74e4b 100644 --- a/Cubical.HITs.RPn.Base.html +++ b/Cubical.HITs.RPn.Base.html @@ -70,8 +70,8 @@ isContrBoolPointedEquiv : ∀ x → isContr ((Bool , false) ≃[ PointedEquivStr ] (Bool , x)) fst (isContrBoolPointedEquiv x) = ((λ y → x ⊕ y) , isEquiv-⊕ x) , ⊕-comm x false snd (isContrBoolPointedEquiv x) (e , p) - = Σ≡Prop (λ e → isSetBool (equivFun e false) x) - (Σ≡Prop isPropIsEquiv (funExt λ { false → ⊕-comm x false ∙ sym p + = Σ≡Prop (λ e → isSetBool (equivFun e false) x) + (Σ≡Prop isPropIsEquiv (funExt λ { false → ⊕-comm x false ∙ sym p ; true → ⊕-comm x true ∙ sym q })) where q : e .fst true ≡ not x q with dichotomyBool (invEq e (not x)) @@ -235,7 +235,7 @@ where open ⊕* (cov⁻¹ (-1+ n) x) i : Total (cov⁻¹ (ℕ→ℕ₋₁ n)) ≃ Pushout Σf Σg - i = (Σ[ x ∈ RP (ℕ→ℕ₋₁ n) ] typ (cov⁻¹ (ℕ→ℕ₋₁ n) x)) ≃⟨ Σ-cong-equiv-snd cov⁻¹≃E ⟩ + i = (Σ[ x ∈ RP (ℕ→ℕ₋₁ n) ] typ (cov⁻¹ (ℕ→ℕ₋₁ n) x)) ≃⟨ Σ-cong-equiv-snd cov⁻¹≃E ⟩ (Σ[ x ∈ RP (ℕ→ℕ₋₁ n) ] E x) ≃⟨ flatten ⟩ Pushout Σf Σg ■ {- @@ -261,9 +261,9 @@ -} u : ∀ {n} → (Σ[ x ∈ Total (cov⁻¹ n) ] typ (cov⁻¹ n (fst x))) ≃ (Total (cov⁻¹ n) × Bool) u {n} = Σ[ x ∈ Total (cov⁻¹ n) ] typ (cov⁻¹ n (fst x)) ≃⟨ Σ-assoc-≃ ⟩ - Σ[ x ∈ RP n ] (typ (cov⁻¹ n x)) × (typ (cov⁻¹ n x)) ≃⟨ Σ-cong-equiv-snd (λ x → Σ-swap-≃) ⟩ - Σ[ x ∈ RP n ] (typ (cov⁻¹ n x)) × (typ (cov⁻¹ n x)) ≃⟨ Σ-cong-equiv-snd - (λ x → Σ-cong-equiv-snd + Σ[ x ∈ RP n ] (typ (cov⁻¹ n x)) × (typ (cov⁻¹ n x)) ≃⟨ Σ-cong-equiv-snd (λ x → Σ-swap-≃) ⟩ + Σ[ x ∈ RP n ] (typ (cov⁻¹ n x)) × (typ (cov⁻¹ n x)) ≃⟨ Σ-cong-equiv-snd + (λ x → Σ-cong-equiv-snd (λ y → ⊕*.Equivˡ (cov⁻¹ n x) y)) ⟩ Σ[ x ∈ RP n ] (typ (cov⁻¹ n x)) × Bool ≃⟨ invEquiv Σ-assoc-≃ ⟩ Total (cov⁻¹ n) × Bool ■ @@ -272,7 +272,7 @@ H x = refl nat : 3-span-equiv (3span Σf Σg) (3span {A2 = Total (cov⁻¹ (-1+ n)) × Bool} fst snd) - nat = record { e0 = idEquiv _ ; e2 = u ; e4 = ΣUnit _ + nat = record { e0 = idEquiv _ ; e2 = u ; e4 = ΣUnit _ ; H1 = λ x → cong fst (H x) ; H3 = λ x → cong snd (H x) } diff --git a/Cubical.HITs.SetQuotients.Properties.html b/Cubical.HITs.SetQuotients.Properties.html index 1f20ac4f2e..21f8edd212 100644 --- a/Cubical.HITs.SetQuotients.Properties.html +++ b/Cubical.HITs.SetQuotients.Properties.html @@ -233,7 +233,7 @@ hright : ∀ a b b' → S b b' → (a ∗ b) ≡ (a ∗ b') hright a _ _ r = h _ _ _ _ (isReflR a) r Iso.rightInv (setQuotUniversal2Iso Bset isReflR isReflS) (_∗_ , h) = - Σ≡Prop (λ _ → isPropΠ4 λ _ _ _ _ → isPropΠ2 λ _ _ → Bset _ _) refl + Σ≡Prop (λ _ → isPropΠ4 λ _ _ _ _ → isPropΠ2 λ _ _ → Bset _ _) refl Iso.leftInv (setQuotUniversal2Iso Bset isReflR isReflS) _∗/_ = funExt₂ (elimProp2 (λ _ _ → Bset _ _) λ _ _ → refl) @@ -278,7 +278,7 @@ rec isSetHProp (λ c → (R a c , Rprop a c)) (λ c d cd → - Σ≡Prop (λ _ → isPropIsProp) + Σ≡Prop (λ _ → isPropIsProp) (hPropExt (Rprop a c) (Rprop a d) (λ ac → R/trans _ _ _ ac cd) (λ ad → R/trans _ _ _ ad (R/sym _ _ cd)))) diff --git a/Cubical.HITs.SetTruncation.Fibers.html b/Cubical.HITs.SetTruncation.Fibers.html index 19460f2d44..e1ebaf1a5d 100644 --- a/Cubical.HITs.SetTruncation.Fibers.html +++ b/Cubical.HITs.SetTruncation.Fibers.html @@ -102,7 +102,7 @@ fiber∥∥₂→∥fiber∥₂/R→fiber∥∥₂ : (x : fiber ∥f∥₂ ∣ y ∣₂) → ∥fiber∥₂/R→fiber∥∥₂ (fiber∥∥₂→∥fiber∥₂/R x) ≡ x fiber∥∥₂→∥fiber∥₂/R→fiber∥∥₂ x = - Σ≡Prop + Σ≡Prop (λ _ → squash₂ _ _) (∥fiber∥₂/R→fiber∥∥₂→fst (fiber∥∥₂→∥fiber∥₂/R x) ∙ fiber∥∥₂→∥fiber∥₂/R→proj x) diff --git a/Cubical.HITs.SmashProduct.Base.html b/Cubical.HITs.SmashProduct.Base.html index 02ba2f783f..0928bc7c25 100644 --- a/Cubical.HITs.SmashProduct.Base.html +++ b/Cubical.HITs.SmashProduct.Base.html @@ -120,9 +120,9 @@ Σ[ l ∈ ((x : fst A) → f x (pt B) ≡ pt C) ] Σ[ r ∈ ((b : fst B) → f (pt A) b ≡ pt C) ] l (pt A) ≡ r (pt B))) - is₂ = compIso is₁ (Σ-cong-iso-snd - λ f → Σ-cong-iso-snd - λ l → Σ-cong-iso-snd + is₂ = compIso is₁ (Σ-cong-iso-snd + λ f → Σ-cong-iso-snd + λ l → Σ-cong-iso-snd λ r → pathToIso (PathP≡doubleCompPathʳ _ _ _ _ ∙ cong (l (snd A) ≡_) (sym (compPath≡compPath' (r (snd B)) refl) @@ -159,12 +159,12 @@ → Iso (Σ _ B) (B (a , refl)) isContrIso a B = compIso (invIso - (Σ-cong-iso-fst (isContr→Iso isContrUnit (isContrSingl a)))) + (Σ-cong-iso-fst (isContr→Iso isContrUnit (isContrSingl a)))) lUnit×Iso iso₄ : Iso (isoToPath is₃ i1) (isoToPath is₂ i1) - iso₄ = Σ-cong-iso-snd λ f → isContrIso (snd C) _ + iso₄ = Σ-cong-iso-snd λ f → isContrIso (snd C) _ ⋀→∙Homogeneous≡ : isHomogeneous C → {f g : (A ⋀∙ B) →∙ C} diff --git a/Cubical.Homotopy.BlakersMassey.html b/Cubical.Homotopy.BlakersMassey.html index a7eb12d975..34695aaf0a 100644 --- a/Cubical.Homotopy.BlakersMassey.html +++ b/Cubical.Homotopy.BlakersMassey.html @@ -709,18 +709,18 @@ (Σ[ a ∈ A ] ((Σ[ c ∈ C ] (g a ≡ c)) × (f a ≡ b))) shuffleFibIso₁ f g b = compIso (invIso Σ-assoc-Iso) - (compIso (Σ-cong-iso-fst Σ-swap-Iso) + (compIso (Σ-cong-iso-fst Σ-swap-Iso) (compIso - (Σ-cong-iso-snd (λ y → Σ-swap-Iso)) + (Σ-cong-iso-snd (λ y → Σ-swap-Iso)) (compIso Σ-assoc-Iso - (Σ-cong-iso-snd λ a → invIso Σ-assoc-Iso)))) + (Σ-cong-iso-snd λ a → invIso Σ-assoc-Iso)))) shuffleFibIso₂ : {ℓ ℓ' ℓ'' : Level} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} (f : A → B) (g : A → C) (x : _) → Iso (Σ[ a ∈ A ] ((Σ[ c ∈ C ] (g a ≡ c)) × (f a ≡ x))) (fiber f x) - shuffleFibIso₂ f g x = Σ-cong-iso-snd - λ a → compIso (Σ-cong-iso-fst + shuffleFibIso₂ f g x = Σ-cong-iso-snd + λ a → compIso (Σ-cong-iso-fst (isContr→Iso (isContrSingl (g a)) isContrUnit)) lUnit×Iso @@ -747,8 +747,8 @@ C-con c = isConnectedRetractFromIso (suc m) (compIso - (compIso (Σ-cong-iso-snd - (λ _ → Σ-cong-iso-snd λ _ → Σ-swap-Iso)) + (compIso (Σ-cong-iso-snd + (λ _ → Σ-cong-iso-snd λ _ → Σ-swap-Iso)) (shuffleFibIso₁ g f c)) (shuffleFibIso₂ g f c)) (con-g c) @@ -780,7 +780,7 @@ TotalPathGen×Iso : Iso (Σ (B × C) PushoutGenPath×) (Σ (B × C) PushoutPath×) TotalPathGen×Iso = - Σ-cong-iso-snd λ x + Σ-cong-iso-snd λ x → congIso (invIso (IsoPushoutPushoutGen f g)) Totalfib×Iso : Iso (Σ (B × C) fib×) A diff --git a/Cubical.Homotopy.Connected.html b/Cubical.Homotopy.Connected.html index 8288780f22..534ee77189 100644 --- a/Cubical.Homotopy.Connected.html +++ b/Cubical.Homotopy.Connected.html @@ -804,13 +804,13 @@ X≃X' a' = (Σ[ x ∈ P (inl a') .fst ] ∀ (b : B) → PathP (λ i → P (push a' b i) .fst) x (k (inr b))) - ≃⟨ invEquiv (Σ-cong-equiv-fst (UnitToType≃ _)) ⟩ + ≃⟨ invEquiv (Σ-cong-equiv-fst (UnitToType≃ _)) ⟩ (Σ[ x' ∈ (Unit → P (inl a') .fst) ] ∀ (b : B) → PathP (λ i → P (push a' b i) .fst) (x' tt) (k (inr b))) - ≃⟨ Σ-cong-equiv-snd (λ x' → equivΠCod (λ b → pathToEquiv (PathP≡Path⁻ _ _ _))) ⟩ + ≃⟨ Σ-cong-equiv-snd (λ x' → equivΠCod (λ b → pathToEquiv (PathP≡Path⁻ _ _ _))) ⟩ (Σ[ x' ∈ (Unit → P (inl a') .fst) ] ∀ (b : B) → x' tt ≡ subst⁻ (λ y → P y .fst) (push a' b) (k (inr b))) - ≃⟨ Σ-cong-equiv-snd (λ x' → funExtEquiv) ⟩ + ≃⟨ Σ-cong-equiv-snd (λ x' → funExtEquiv) ⟩ (Σ[ x' ∈ (Unit → P (inl a') .fst) ] (λ (b : B) → x' tt) ≡ (λ (b : B) → subst⁻ (λ y → P y .fst) (push a' b) (k (inr b)))) diff --git a/Cubical.Homotopy.EilenbergMacLane.CupProduct.html b/Cubical.Homotopy.EilenbergMacLane.CupProduct.html index c54767c434..fd87da4e53 100644 --- a/Cubical.Homotopy.EilenbergMacLane.CupProduct.html +++ b/Cubical.Homotopy.EilenbergMacLane.CupProduct.html @@ -134,7 +134,7 @@ where lem : compGroupHom (GroupEquiv→GroupHom ⨂assoc) TensorMult3Homₗ ≡ compGroupHom (inducedHom⨂ idGroupHom TensorMultHom) TensorMultHom - lem = Σ≡Prop (λ _ → isPropIsGroupHom _ _) + lem = Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (⊗elimProp (λ _ → is-set (snd G') _ _) (λ a → ⊗elimProp (λ _ → is-set (snd G') _ _) (λ b c → sym (·Assoc (snd G'') a b c)) @@ -195,7 +195,7 @@ (TensorMultHom {G' = CommRing→Ring G''}) ≡ TensorMultHom isTrivComm = - Σ≡Prop (λ _ → isPropIsGroupHom _ _) + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (⊗elimProp (λ _ → CommRingStr.is-set (snd G'') _ _) (λ a b → CommRingStr.·Comm (snd G'') b a) λ p q r s → cong₂ _+G_ r s)) diff --git a/Cubical.Homotopy.EilenbergMacLane.GradedCommTensor.html b/Cubical.Homotopy.EilenbergMacLane.GradedCommTensor.html index e094eff623..895102e91b 100644 --- a/Cubical.Homotopy.EilenbergMacLane.GradedCommTensor.html +++ b/Cubical.Homotopy.EilenbergMacLane.GradedCommTensor.html @@ -461,7 +461,7 @@ h : Path (AbGroupHom (H' ⨂ G') (G' ⨂ H')) (GroupEquiv→GroupHom ⨂-comm) (GroupEquiv→GroupHom (invGroupEquiv ⨂-comm)) - h = Σ≡Prop (λ _ → isPropIsGroupHom _ _) refl + h = Σ≡Prop (λ _ → isPropIsGroupHom _ _) refl -- cong² comm⨂-EM commutes with wrap cong-cong-comm⨂-EM : (n : ℕ) (p : fst (Ω (EM∙ (H' ⨂ G') (suc (suc n))))) diff --git a/Cubical.Homotopy.EilenbergMacLane.Order2.html b/Cubical.Homotopy.EilenbergMacLane.Order2.html index 51f4cd98eb..3ea2180ed1 100644 --- a/Cubical.Homotopy.EilenbergMacLane.Order2.html +++ b/Cubical.Homotopy.EilenbergMacLane.Order2.html @@ -111,7 +111,7 @@ → PathP (λ i → Path (EM G 1) embase (emloop g i) → TypeOfHLevel ℓ 1) (λ p → (p ≡ sym p) , hLevelEM G 1 _ _ _ _) λ p → (p ≡ sym p) , hLevelEM G 1 _ _ _ _ - main g = toPathP (funExt (λ p → Σ≡Prop (λ _ → isPropIsProp) + main g = toPathP (funExt (λ p → Σ≡Prop (λ _ → isPropIsProp) (funExt⁻ (fromPathP (main' g)) p))) symCode (suc n) = TR.elim (λ _ → isOfHLevelΠ (4 +ℕ n) diff --git a/Cubical.Homotopy.EilenbergMacLane.Properties.html b/Cubical.Homotopy.EilenbergMacLane.Properties.html index 6141954641..f208453970 100644 --- a/Cubical.Homotopy.EilenbergMacLane.Properties.html +++ b/Cubical.Homotopy.EilenbergMacLane.Properties.html @@ -133,7 +133,7 @@ CodesSet = EMrec Ĝ (isOfHLevelTypeOfHLevel 2) (G , is-set) RE REComp where RE : (g : G) → Path (hSet ℓ) (G , is-set) (G , is-set) - RE g = Σ≡Prop (λ X → isPropIsOfHLevel {A = X} 2) (ua (rightEquiv g)) + RE g = Σ≡Prop (λ X → isPropIsOfHLevel {A = X} 2) (ua (rightEquiv g)) lemma₁ : (g h : G) → Square (ua (rightEquiv g)) (ua (rightEquiv (g · h))) @@ -197,9 +197,9 @@ → Path (TypeOfHLevel _ (3 + n)) (EM G (suc n) , hLevelEM G (suc n)) (EM G (suc n) , hLevelEM G (suc n)) - fib zero a = Σ≡Prop (λ _ → isPropIsOfHLevel 3) + fib zero a = Σ≡Prop (λ _ → isPropIsOfHLevel 3) (isoToPath (addIso 1 a)) - fib (suc n) a = Σ≡Prop (λ _ → isPropIsOfHLevel (4 + n)) + fib (suc n) a = Σ≡Prop (λ _ → isPropIsOfHLevel (4 + n)) (isoToPath (addIso (suc (suc n)) ∣ a ∣)) decode' : (n : ℕ) (x : EM G (suc (suc n))) → CODE n x .fst → 0ₖ (suc (suc n)) ≡ x @@ -521,7 +521,7 @@ isContr-↓∙ : {G : AbGroup ℓ} {H : AbGroup ℓ'} (n : ℕ) → isContr (EM∙ G (suc n) →∙ EM∙ H n) fst (isContr-↓∙ {G = G} {H = H} zero) = (λ _ → 0g (snd H)) , refl snd (isContr-↓∙{G = G} {H = H} zero) (f , p) = - Σ≡Prop (λ x → is-set (snd H) _ _) + Σ≡Prop (λ x → is-set (snd H) _ _) (funExt (raw-elim G 0 (λ _ → is-set (snd H) _ _) (sym p))) fst (isContr-↓∙ {G = G} {H = H} (suc n)) = (λ _ → 0ₖ (suc n)) , refl fst (snd (isContr-↓∙ {G = G} {H = H} (suc n)) f i) x = diff --git a/Cubical.Homotopy.Group.Base.html b/Cubical.Homotopy.Group.Base.html index 2e1500008b..78a5539410 100644 --- a/Cubical.Homotopy.Group.Base.html +++ b/Cubical.Homotopy.Group.Base.html @@ -1029,7 +1029,7 @@ invEquiv∙idEquiv∙≡idEquiv : ∀ {ℓ} {A : Pointed ℓ} → invEquiv∙ (idEquiv (fst A) , (λ _ → pt A)) ≡ (idEquiv (fst A) , refl) -invEquiv∙idEquiv∙≡idEquiv = ΣPathP ((Σ≡Prop (λ _ → isPropIsEquiv _) refl) , (sym (lUnit refl))) +invEquiv∙idEquiv∙≡idEquiv = ΣPathP ((Σ≡Prop (λ _ → isPropIsEquiv _) refl) , (sym (lUnit refl))) π'eqFunIsEquiv : ∀ {ℓ} {A : Pointed ℓ} {B : Pointed ℓ} (n : ℕ) diff --git a/Cubical.Homotopy.Group.LES.html b/Cubical.Homotopy.Group.LES.html index d0cdb5a744..f5ef8f0eef 100644 --- a/Cubical.Homotopy.Group.LES.html +++ b/Cubical.Homotopy.Group.LES.html @@ -649,14 +649,14 @@ (πLES.A→B f n) (π'∘∙Hom n f) π∘∙A→B-PathP n f = - toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) (π'∘∙Hom'≡π'∘∙fun n f)) + toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) (π'∘∙Hom'≡π'∘∙fun n f)) π∘∙fib→A-PathP : ∀ {ℓ ℓ'} {A : Pointed ℓ} {B : Pointed ℓ'} (n : ℕ) (f : A →∙ B) → PathP (λ i → GroupHomπ≅π'PathP (ΩLES.fibf f) A n i) (πLES.fib→A f n) (π'∘∙Hom n (fst , refl)) π∘∙fib→A-PathP {A = A} {B = B} n f = - toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) + toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) (cong (transport (λ i → (fst (GroupPath _ _) (GroupIso→GroupEquiv (π'Gr≅πGr n (ΩLES.fibf f))) (~ i)) .fst diff --git a/Cubical.Homotopy.Group.Pi4S3.BrunerieNumber.html b/Cubical.Homotopy.Group.Pi4S3.BrunerieNumber.html index 38363c3103..2f756b4688 100644 --- a/Cubical.Homotopy.Group.Pi4S3.BrunerieNumber.html +++ b/Cubical.Homotopy.Group.Pi4S3.BrunerieNumber.html @@ -165,8 +165,8 @@ fiberinr'Iso' : Iso (fiber inr' (inl tt)) (Σ (Unit × S₊ 2) PushoutPath×) fiberinr'Iso' = - compIso (Σ-cong-iso-snd (λ x → symIso)) - (Σ-cong-iso-fst (invIso lUnit×Iso)) + compIso (Σ-cong-iso-snd (λ x → symIso)) + (Σ-cong-iso-fst (invIso lUnit×Iso)) fiberinr'Iso : Iso (fiber inr' (inl tt)) (Σ (Unit × S₊ 2) PushoutPath×) @@ -257,9 +257,9 @@ (sym (GroupPath _ _ .fst π₂P≅0)) _ _ _ _ _ P→S²→Pushout - (ΣPathPProp (λ _ → isPropIsGroupHom _ _) + (ΣPathPProp (λ _ → isPropIsGroupHom _ _) λ i → fst (π∘∙fib→A-PathP 2 inr∙ i)) - (ΣPathPProp (λ _ → isPropIsGroupHom _ _) + (ΣPathPProp (λ _ → isPropIsGroupHom _ _) λ i → fst (π∘∙A→B-PathP 2 inr∙ i)) -- The two surjections in question @@ -291,7 +291,7 @@ transportLem : PathP (λ i → GroupHomπ≅π'PathP (S₊∙ 3) TotalPushoutPath×∙ 2 i) π₃S³→π₃TotalPushoutPath× π₃S³→π₃P transportLem = - toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) + toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) (π'∘∙Hom'≡π'∘∙fun {A = S₊∙ 3} {B = TotalPushoutPath×∙} 2 (S³→TotalPushoutPath× , refl))) @@ -322,7 +322,7 @@ (π'∘∙Hom 2 TotalPushoutPath×∙→P) (π'∘∙Hom 2 (fst , refl)))) ≡ π'∘∙Hom 2 (fold∘W , refl) tripleComp≡ = - Σ≡Prop (λ _ → isPropIsGroupHom _ _) + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (sElim (λ _ → isSetPathImplicit) λ f → cong ∣_∣₂ (ΣPathP (refl , (cong (_∙ refl) (λ j → cong fst (rUnit (cong (fst TotalPushoutPath×∙→P) diff --git a/Cubical.Homotopy.Group.Pi4S3.DirectProof.html b/Cubical.Homotopy.Group.Pi4S3.DirectProof.html index c756899758..d10d888067 100644 --- a/Cubical.Homotopy.Group.Pi4S3.DirectProof.html +++ b/Cubical.Homotopy.Group.Pi4S3.DirectProof.html @@ -466,7 +466,7 @@ π₃*joinS¹S¹→π₃*S²' (fst (fst GrEq) , snd GrEq) help = - toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) + toPathP (Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ f → (λ i → transportRefl diff --git a/Cubical.Homotopy.HopfInvariant.Brunerie.html b/Cubical.Homotopy.HopfInvariant.Brunerie.html index ce5079e63d..f2f6bd647b 100644 --- a/Cubical.Homotopy.HopfInvariant.Brunerie.html +++ b/Cubical.Homotopy.HopfInvariant.Brunerie.html @@ -178,7 +178,7 @@ (compIso (fst (Hⁿ-⋁ (S₊∙ 2) (S₊∙ 2) 2)) (compIso - (prodIso (fst (Hⁿ-Sᵐ≅0 2 1 λ p → snotz (cong predℕ p))) + (prodIso (fst (Hⁿ-Sᵐ≅0 2 1 λ p → snotz (cong predℕ p))) (fst (Hⁿ-Sᵐ≅0 2 1 λ p → snotz (cong predℕ p)))) rUnit×Iso)) isContrUnit)) @@ -187,7 +187,7 @@ (compIso (fst (Hⁿ-⋁ (S₊∙ 2) (S₊∙ 2) 3)) (compIso - (prodIso (fst (Hⁿ-Sᵐ≅0 3 1 λ p → snotz (cong predℕ p))) + (prodIso (fst (Hⁿ-Sᵐ≅0 3 1 λ p → snotz (cong predℕ p))) (fst (Hⁿ-Sᵐ≅0 3 1 λ p → snotz (cong predℕ p)))) rUnit×Iso)) isContrUnit))) diff --git a/Cubical.Homotopy.Whitehead.html b/Cubical.Homotopy.Whitehead.html index da71e9b58f..39095e54af 100644 --- a/Cubical.Homotopy.Whitehead.html +++ b/Cubical.Homotopy.Whitehead.html @@ -373,7 +373,7 @@ -- Main iso Iso-Susp×Susp-cofibJoinTo⋁ : Iso (Susp A × Susp B) cofibW Iso-Susp×Susp-cofibJoinTo⋁ = - compIso (Σ-cong-iso-snd (λ _ → invSuspIso)) + compIso (Σ-cong-iso-snd (λ _ → invSuspIso)) Iso₁-Susp×Susp-cofibW -- The induced function A ∨ B → Susp A × Susp B satisfies @@ -391,7 +391,7 @@ f₁ = fun Iso-PushSusp×-Susp×Susp f₂ = fun Iso-A□○-PushSusp× f₃ = backward-l whitehead3x3 - f₄ = fun (Σ-cong-iso-snd (λ _ → invSuspIso)) + f₄ = fun (Σ-cong-iso-snd (λ _ → invSuspIso)) lem : (b : B) → cong (f₁ ∘ f₂ ∘ f₃) (push b) diff --git a/Cubical.Papers.AffineSchemes.html b/Cubical.Papers.AffineSchemes.html index 93bc66381b..439788ebf6 100644 --- a/Cubical.Papers.AffineSchemes.html +++ b/Cubical.Papers.AffineSchemes.html @@ -183,7 +183,7 @@ ---------- 4: Category Theory ---------- -- Σ-subcategories -open CatTheory using (ΣPropCat) +open CatTheory using (ΣPropCat) -- Kan-extension for distributive lattices open SheafExtension using (DLRan ; DLRanNatIso) diff --git a/Cubical.Relation.Nullary.DecidablePropositions.html b/Cubical.Relation.Nullary.DecidablePropositions.html index 3432a2831b..156aac163a 100644 --- a/Cubical.Relation.Nullary.DecidablePropositions.html +++ b/Cubical.Relation.Nullary.DecidablePropositions.html @@ -42,7 +42,7 @@ isPropIsDecProp : {P : Type ℓ} → isProp (isDecProp P) isPropIsDecProp p q = - Σ≡PropEquiv (λ _ → isOfHLevel⁺≃ᵣ 0 isPropBool→Type) .fst + Σ≡PropEquiv (λ _ → isOfHLevel⁺≃ᵣ 0 isPropBool→Type) .fst (Bool→TypeInj _ _ (invEquiv (p .snd) ⋆ q .snd)) isDecPropRespectEquiv : {P : Type ℓ} {Q : Type ℓ'} diff --git a/Cubical.Structures.Axioms.html b/Cubical.Structures.Axioms.html index 3e4e23ad75..2ec96dc9ca 100644 --- a/Cubical.Structures.Axioms.html +++ b/Cubical.Structures.Axioms.html @@ -43,7 +43,7 @@ ι (X , s) (Y , t) e ≃⟨ θ e ⟩ PathP (λ i → S (ua e i)) s t - ≃⟨ invEquiv (Σ-contractSnd λ _ → isOfHLevelPathP' 0 (axioms-are-Props _ _) _ _) ⟩ + ≃⟨ invEquiv (Σ-contractSnd λ _ → isOfHLevelPathP' 0 (axioms-are-Props _ _) _ _) ⟩ Σ[ p ∈ PathP (λ i → S (ua e i)) s t ] PathP (λ i → axioms (ua e i) (p i)) a b ≃⟨ ΣPath≃PathΣ ⟩ PathP (λ i → AxiomsStructure S axioms (ua e i)) (s , a) (t , b) diff --git a/Cubical.Structures.Product.html b/Cubical.Structures.Product.html index 762c41a888..22150626dc 100644 --- a/Cubical.Structures.Product.html +++ b/Cubical.Structures.Product.html @@ -36,13 +36,13 @@ {S₂ : Type ℓ → Type ℓ₂} (ι₂ : StrEquiv S₂ ℓ₂') (θ₂ : UnivalentStr S₂ ι₂) → UnivalentStr (ProductStructure S₁ S₂) (ProductEquivStr ι₁ ι₂) productUnivalentStr {S₁ = S₁} ι₁ θ₁ {S₂} ι₂ θ₂ {X , s₁ , s₂} {Y , t₁ , t₂} e = - compEquiv (Σ-cong-equiv (θ₁ e) (λ _ → θ₂ e)) ΣPath≃PathΣ + compEquiv (Σ-cong-equiv (θ₁ e) (λ _ → θ₂ e)) ΣPath≃PathΣ productEquivAction : {S₁ : Type ℓ → Type ℓ₁} (α₁ : EquivAction S₁) {S₂ : Type ℓ → Type ℓ₂} (α₂ : EquivAction S₂) → EquivAction (ProductStructure S₁ S₂) -productEquivAction α₁ α₂ e = Σ-cong-equiv (α₁ e) (λ _ → α₂ e) +productEquivAction α₁ α₂ e = Σ-cong-equiv (α₁ e) (λ _ → α₂ e) productTransportStr : {S₁ : Type ℓ → Type ℓ₁} (α₁ : EquivAction S₁) (τ₁ : TransportStr α₁) diff --git a/Cubical.Structures.Relational.Equalizer.html b/Cubical.Structures.Relational.Equalizer.html index 53c7d09e86..f325492086 100644 --- a/Cubical.Structures.Relational.Equalizer.html +++ b/Cubical.Structures.Relational.Equalizer.html @@ -40,8 +40,8 @@ equalizerSuitableRel {f = f} {g} {ρ₁} {ρ₂} αf αg θ₁ θ₂ .quo (X , s , p) (R , qer) r = ( ((quo₁ .fst .fst , sym step₀ ∙ step₁) , quo₁ .fst .snd) , λ {((s' , _) , r') → - Σ≡Prop (λ _ → θ₁ .prop (λ _ _ → squash/ _ _) _ _) - (Σ≡Prop (λ _ → θ₂ .set squash/ _ _) + Σ≡Prop (λ _ → θ₁ .prop (λ _ _ → squash/ _ _) _ _) + (Σ≡Prop (λ _ → θ₂ .set squash/ _ _) (cong fst (quo₁ .snd (s' , r'))))} ) where diff --git a/Cubical.Structures.Relational.Function.html b/Cubical.Structures.Relational.Function.html index 3d01c71b34..2751f89d51 100644 --- a/Cubical.Structures.Relational.Function.html +++ b/Cubical.Structures.Relational.Function.html @@ -120,7 +120,7 @@ (invIsEq (σ₁ .quo R) t) (subst (ρ₁ (graphRel [_]) s) (sym (secIsEq (σ₁ .quo R) t)) r) final .snd (f' , c) = - Σ≡Prop + Σ≡Prop (λ _ → isPropImplicitΠ λ s → isPropImplicitΠ λ t → isPropΠ λ _ → θ₂ .prop (λ _ _ → squash/ _ _) _ _) diff --git a/Cubical.Structures.Relational.Product.html b/Cubical.Structures.Relational.Product.html index 7bc964766c..418ba49153 100644 --- a/Cubical.Structures.Relational.Product.html +++ b/Cubical.Structures.Relational.Product.html @@ -63,7 +63,7 @@ → StrRelMatchesEquiv ρ₁ ι₁ → StrRelMatchesEquiv ρ₂ ι₂ → StrRelMatchesEquiv (ProductRelStr ρ₁ ρ₂) (ProductEquivStr ι₁ ι₂) productRelMatchesEquiv ρ₁ ρ₂ μ₁ μ₂ A B e = - Σ-cong-equiv (μ₁ _ _ e) (λ _ → μ₂ _ _ e) + Σ-cong-equiv (μ₁ _ _ e) (λ _ → μ₂ _ _ e) productRelAction : {S₁ : Type ℓ → Type ℓ₁} {ρ₁ : StrRel S₁ ℓ₁'} (α₁ : StrRelAction ρ₁) @@ -93,7 +93,7 @@ (funExt (elimProp (λ _ → productSuitableRel θ₁ θ₂ .set squash/ _ _) (λ _ → refl))) (compEquiv (isoToEquiv isom) - (Σ-cong-equiv (_ , σ₁ .quo R) (λ _ → _ , σ₂ .quo R)) .snd) + (Σ-cong-equiv (_ , σ₁ .quo R) (λ _ → _ , σ₂ .quo R)) .snd) where fwd : ProductStructure S₁ S₂ X / ProductRelStr ρ₁ ρ₂ (R .fst .fst) @@ -138,5 +138,5 @@ → StrRelMatchesEquiv ρ₂ (EquivAction→StrEquiv α₂) → StrRelMatchesEquiv (ProductRelStr ρ₁ ρ₂) (EquivAction→StrEquiv (productEquivAction α₁ α₂)) productRelMatchesTransp _ _ _ _ μ₁ μ₂ _ _ e = - compEquiv (Σ-cong-equiv (μ₁ _ _ e) (λ _ → μ₂ _ _ e)) ΣPath≃PathΣ + compEquiv (Σ-cong-equiv (μ₁ _ _ e) (λ _ → μ₂ _ _ e)) ΣPath≃PathΣ \ No newline at end of file diff --git a/Cubical.Tactics.CategorySolver.Solver.html b/Cubical.Tactics.CategorySolver.Solver.html index bd1984f7dd..acc7b8949e 100644 --- a/Cubical.Tactics.CategorySolver.Solver.html +++ b/Cubical.Tactics.CategorySolver.Solver.html @@ -15,20 +15,20 @@ variable ℓ ℓ' : Level open Category -open Functor +open Functor module Eval (𝓒 : Category ℓ ℓ') where -- Semantics in 𝓒 itself, tautologically open FreeCategory (Cat→Graph 𝓒) sem𝓒 = ε {𝓒 = 𝓒} - ⟦_⟧c = sem𝓒 .F-hom + ⟦_⟧c = sem𝓒 .F-hom 𝓟 = PowerCategory (𝓒 .ob) (SET (ℓ-max ℓ ℓ')) - 𝓘 : Functor FreeCat 𝓟 + 𝓘 : Functor FreeCat 𝓟 𝓘 = PseudoYoneda {C = FreeCat} -- Semantics in 𝓟 (𝓒 .ob), interpreting fun symbols using Yoneda module YoSem = Semantics 𝓟 (𝓘 ∘Interp η) - ⟦_⟧yo = YoSem.sem .F-hom + ⟦_⟧yo = YoSem.sem .F-hom -- | Evaluate by taking the semantics in 𝓟 (𝓒 .ob) eval : ∀ {A B} → FreeCat [ A , B ] → _ @@ -45,9 +45,9 @@ → eval e₁ ≡ eval e₂ → ⟦ e₁ ⟧c ≡ ⟦ e₂ ⟧c solve {A}{B} e₁ e₂ p = cong ⟦_⟧c (isFaithfulPseudoYoneda _ _ _ _ lem) where - lem : 𝓘 ⟪ e₁ ⟫ ≡ 𝓘 ⟪ e₂ ⟫ + lem : 𝓘 ⟪ e₁ ⟫ ≡ 𝓘 ⟪ e₂ ⟫ lem = transport - (λ i → Yo-YoSem-agree (~ i) ⟪ e₁ ⟫ ≡ Yo-YoSem-agree (~ i) ⟪ e₂ ⟫) + (λ i → Yo-YoSem-agree (~ i) ⟪ e₁ ⟫ ≡ Yo-YoSem-agree (~ i) ⟪ e₂ ⟫) p solve : (𝓒 : Category ℓ ℓ') diff --git a/Cubical.Tactics.FunctorSolver.Examples.html b/Cubical.Tactics.FunctorSolver.Examples.html index d0f2e8ede1..ba1cd1ec17 100644 --- a/Cubical.Tactics.FunctorSolver.Examples.html +++ b/Cubical.Tactics.FunctorSolver.Examples.html @@ -21,18 +21,18 @@ variable ℓ ℓ' : Level C D : Category ℓ ℓ' - F : Functor C D + F : Functor C D -module Examples (F : Functor C D) where +module Examples (F : Functor C D) where open Category - open Functor + open Functor _ : ∀ {A B}{f : D [ A , B ]} → D .id ∘⟨ D ⟩ f ≡ f ∘⟨ D ⟩ D .id _ = solveFunctor! C D F _ : ∀ {A} - → D .id ≡ F ⟪ (C .id {A}) ⟫ + → D .id ≡ F ⟪ (C .id {A}) ⟫ _ = solveFunctor! C D F @@ -40,8 +40,8 @@ → {Z : D .ob} → {f : C [ W , X ]} → {g : C [ X , Y ]} - → {h : D [ F ⟅ Y ⟆ , Z ]} - → h ∘⟨ D ⟩ F ⟪ (g ∘⟨ C ⟩ C .id) ∘⟨ C ⟩ f ⟫ ∘⟨ D ⟩ F ⟪ C .id ⟫ - ≡ D .id ∘⟨ D ⟩ h ∘⟨ D ⟩ F ⟪ C .id ∘⟨ C ⟩ g ⟫ ∘⟨ D ⟩ F ⟪ f ⟫ + → {h : D [ F ⟅ Y ⟆ , Z ]} + → h ∘⟨ D ⟩ F ⟪ (g ∘⟨ C ⟩ C .id) ∘⟨ C ⟩ f ⟫ ∘⟨ D ⟩ F ⟪ C .id ⟫ + ≡ D .id ∘⟨ D ⟩ h ∘⟨ D ⟩ F ⟪ C .id ∘⟨ C ⟩ g ⟫ ∘⟨ D ⟩ F ⟪ f ⟫ _ = solveFunctor! C D F \ No newline at end of file diff --git a/Cubical.Tactics.FunctorSolver.Reflection.html b/Cubical.Tactics.FunctorSolver.Reflection.html index 423cb5178e..84558bfaa7 100644 --- a/Cubical.Tactics.FunctorSolver.Reflection.html +++ b/Cubical.Tactics.FunctorSolver.Reflection.html @@ -44,7 +44,7 @@ def (quote Category._⋆_) (category-args (_ h∷ _ h∷ _ h∷ f v∷ g v∷ [])) pattern “F” functor f = - def (quote Functor.F-hom) (functor-args functor (_ h∷ _ h∷ f v∷ [])) + def (quote Functor.F-hom) (functor-args functor (_ h∷ _ h∷ f v∷ [])) -- Parse the input into an exp buildDomExpression : Term → Term @@ -75,7 +75,7 @@ → TC Unit solve-macro b dom cod fctor = equation-solver - (quote Category.id ∷ quote Category._⋆_ ∷ quote Functor.F-hom ∷ []) + (quote Category.id ∷ quote Category._⋆_ ∷ quote Functor.F-hom ∷ []) mk-call b where diff --git a/Cubical.Tactics.FunctorSolver.Solver.html b/Cubical.Tactics.FunctorSolver.Solver.html index 57a14b972c..5e4fa48abb 100644 --- a/Cubical.Tactics.FunctorSolver.Solver.html +++ b/Cubical.Tactics.FunctorSolver.Solver.html @@ -15,7 +15,7 @@ open import Cubical.Categories.Category open import Cubical.Categories.Constructions.Free.Functor open import Cubical.Categories.Constructions.Power -open import Cubical.Categories.Functor renaming (Id to IdF) +open import Cubical.Categories.Functor renaming (Id to IdF) open import Cubical.Categories.Instances.Sets open import Cubical.Categories.NaturalTransformation open import Cubical.Categories.UnderlyingGraph @@ -24,12 +24,12 @@ variable ℓc ℓc' ℓd ℓd' ℓb ℓb' : Level open Category -open Functor +open Functor open NatIso open NatTrans -module Eval (𝓒 : Category ℓc ℓc') (𝓓 : Category ℓd ℓd') (𝓕 : Functor 𝓒 𝓓) where - open FreeFunctor (Cat→Graph 𝓒) (Cat→Graph 𝓓) (𝓕 .F-ob) +module Eval (𝓒 : Category ℓc ℓc') (𝓓 : Category ℓd ℓd') (𝓕 : Functor 𝓒 𝓓) where + open FreeFunctor (Cat→Graph 𝓒) (Cat→Graph 𝓓) (𝓕 .F-ob) Free𝓒 = FG η𝓒 = ηG @@ -37,31 +37,31 @@ η𝓓 = ηH Free𝓕 = Fϕ 𝓟 = PowerCategory (Free𝓓 .ob) (SET (ℓ-max (ℓ-max (ℓ-max ℓc ℓc') ℓd) ℓd')) - PsYo : Functor Free𝓓 𝓟 + PsYo : Functor Free𝓓 𝓟 PsYo = PseudoYoneda {C = Free𝓓} module TautoSem = Semantics {𝓒 = 𝓒} {𝓓 = 𝓓} {𝓕 = 𝓕} IdHom IdHom reflEq module YoSem = Semantics {𝓒 = 𝓟} {𝓓 = 𝓟} {𝓕 = IdF} - (Functor→GraphHom (PsYo ∘F Free𝓕) ∘GrHom η𝓒) + (Functor→GraphHom (PsYo ∘F Free𝓕) ∘GrHom η𝓒) (Functor→GraphHom PsYo ∘GrHom η𝓓) reflEq Yo-YoSem-Agree : Path _ PsYo YoSem.semH Yo-YoSem-Agree = sem-uniq-H where - open YoSem.Uniqueness (PsYo ∘F Free𝓕) PsYo F-rUnit refl refl + open YoSem.Uniqueness (PsYo ∘F Free𝓕) PsYo F-rUnit refl refl (compPath→Square (sym (lUnit (eqToPath reflEq)) ∙ rUnit refl)) solve : ∀ {A B} → (e e' : Free𝓓 [ A , B ]) - → (p : Path _ (YoSem.semH ⟪ e ⟫) (YoSem.semH ⟪ e' ⟫)) - → Path _ (TautoSem.semH ⟪ e ⟫) (TautoSem.semH ⟪ e' ⟫) + → (p : Path _ (YoSem.semH ⟪ e ⟫) (YoSem.semH ⟪ e' ⟫)) + → Path _ (TautoSem.semH ⟪ e ⟫) (TautoSem.semH ⟪ e' ⟫) solve {A}{B} e e' p = - cong (TautoSem.semH .F-hom) (isFaithfulPseudoYoneda _ _ _ _ lem) where - lem : Path _ (PsYo ⟪ e ⟫) (PsYo ⟪ e' ⟫) + cong (TautoSem.semH .F-hom) (isFaithfulPseudoYoneda _ _ _ _ lem) where + lem : Path _ (PsYo ⟪ e ⟫) (PsYo ⟪ e' ⟫) lem = transport (λ i → Path _ - (Yo-YoSem-Agree (~ i) ⟪ e ⟫) - (Yo-YoSem-Agree (~ i) ⟪ e' ⟫)) + (Yo-YoSem-Agree (~ i) ⟪ e ⟫) + (Yo-YoSem-Agree (~ i) ⟪ e' ⟫)) p solve = Eval.solve diff --git a/Cubical.ZCohomology.EilenbergSteenrodZ.html b/Cubical.ZCohomology.EilenbergSteenrodZ.html index b6f09b251c..4a193fd247 100644 --- a/Cubical.ZCohomology.EilenbergSteenrodZ.html +++ b/Cubical.ZCohomology.EilenbergSteenrodZ.html @@ -102,7 +102,7 @@ snd (H0-susp {A = A}) = ST.elim (λ _ → isOfHLevelPath 2 isSetSetTrunc _ _) λ {(f , p) - → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) + → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) (funExt λ {north → sym p ; south → sym p ∙ cong f (merid (pt A)) ; (merid a i) j → isSet→isSet' (isSetℤ) @@ -192,7 +192,7 @@ rightInv (suspFunCharac0 {A = A}) = ST.elim (λ _ → isOfHLevelPath 2 isSetSetTrunc _ _) λ {(f , p) - → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) + → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) (funExt (λ x → (λ j → transp (λ i → helix (wedgeMapS¹ (intLoop (p j) (~ i)) base)) j ((transport (λ i → helix (T.rec isGroupoidS¹ (λ x → x) (rUnitₖ 1 ∣ intLoop (f x) i ∣ j))) @@ -211,7 +211,7 @@ snd (theMorph (pos zero) f) = makeIsGroupHom (ST.elim2 (λ _ _ → isOfHLevelPath 2 isSetSetTrunc _ _) - λ f g → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl)) + λ f g → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl)) theMorph (pos (suc n)) f = coHomMorph _ (fst f) fst (theMorph (negsuc n) f) = idfun _ snd (theMorph (negsuc n) f) = makeIsGroupHom λ _ _ → refl @@ -284,7 +284,7 @@ (λ gId → ∣ ∣ (λ { (inl tt) → 0 ; (inr b) → g b ; (push a i) → funExt⁻ (cong fst gId) a (~ i)}) , q ∣₂ - , cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl) ∣₁) + , cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl) ∣₁) (Iso.fun PathIdTrunc₀Iso inker)}) inv (exactnessIso (pos zero) (f , p)) = uncurry (ST.elim (λ _ → isSetΠ λ _ → isSetΣ isSetSetTrunc λ _ → isOfHLevelPath 2 isSetSetTrunc _ _) @@ -296,7 +296,7 @@ (λ pushmap pushId' → PT.rec (isSetSetTrunc _ _) (λ pushId - → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) + → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) (funExt λ x → sym (funExt⁻ (cong fst pushId) (f x)) ∙∙ cong (fst pushmap) (sym (push x) ∙ push (pt A)) ∙∙ (cong (fst pushmap ∘ inr) p @@ -307,12 +307,12 @@ uncurry (ST.elim (λ _ → isSetΠ λ _ → isOfHLevelPath 2 (isSetΣ isSetSetTrunc (λ _ → isProp→isSet isPropPropTrunc)) _ _) - λ {(p , q) _ → Σ≡Prop (λ _ → isPropPropTrunc) refl}) + λ {(p , q) _ → Σ≡Prop (λ _ → isPropPropTrunc) refl}) leftInv (exactnessIso (pos zero) (f , p)) = uncurry (ST.elim (λ _ → isSetΠ λ _ → isOfHLevelPath 2 (isSetΣ isSetSetTrunc (λ _ → isProp→isSet (isSetSetTrunc _ _))) _ _) - λ {(p , q) _ → Σ≡Prop (λ _ → isSetSetTrunc _ _) refl}) + λ {(p , q) _ → Σ≡Prop (λ _ → isSetSetTrunc _ _) refl}) fun (exactnessIso (pos (suc n)) f) ker = (fst ker) , inIm-helper (fst ker) (snd ker) where inIm-helper : (x : coHom (suc n) (typ B)) @@ -345,14 +345,14 @@ (λ p → (cong ∣_∣₂ (funExt λ x → cong cg (sym (push x)) ∙ p))) (isConnectedPathKn _ (cg (inl tt)) (0ₖ (suc n)) .fst) - rightInv (exactnessIso (pos (suc n)) f) _ = Σ≡Prop (λ _ → isPropPropTrunc) refl - leftInv (exactnessIso (pos (suc n)) f) _ = Σ≡Prop (λ _ → isSetSetTrunc _ _) refl + rightInv (exactnessIso (pos (suc n)) f) _ = Σ≡Prop (λ _ → isPropPropTrunc) refl + leftInv (exactnessIso (pos (suc n)) f) _ = Σ≡Prop (λ _ → isSetSetTrunc _ _) refl exactnessIso (negsuc n) (f , p) = isContr→Iso ((tt* , refl) - , λ {(tt* , p) → Σ≡Prop (λ _ → isOfHLevelPath 1 isPropUnit* _ _) + , λ {(tt* , p) → Σ≡Prop (λ _ → isOfHLevelPath 1 isPropUnit* _ _) refl}) ((tt* , ∣ tt* , refl ∣₁) - , λ {(tt* , p) → Σ≡Prop (λ _ → isPropPropTrunc) + , λ {(tt* , p) → Σ≡Prop (λ _ → isPropPropTrunc) refl}) -------------------------- Dimension --------------------------- diff --git a/Cubical.ZCohomology.GroupStructure.html b/Cubical.ZCohomology.GroupStructure.html index ff78d63701..0648790752 100644 --- a/Cubical.ZCohomology.GroupStructure.html +++ b/Cubical.ZCohomology.GroupStructure.html @@ -526,7 +526,7 @@ commₕ∙ zero = ST.elim2 (λ _ _ → isOfHLevelPath 2 § _ _) λ {(f , p) (g , q) - → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → commₖ 0 (f x) (g x) i)} + → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → commₖ 0 (f x) (g x) i)} commₕ∙ (suc zero) = ST.elim2 (λ _ _ → isOfHLevelPath 2 § _ _) λ {(f , p) (g , q) @@ -545,7 +545,7 @@ rUnitₕ∙ : {A : Pointed ℓ} (n : ℕ) (x : coHomRed n A) → x +[ n ]ₕ∙ 0ₕ∙ n ≡ x rUnitₕ∙ zero = ST.elim (λ _ → isOfHLevelPath 2 § _ _) - λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → rUnitₖ zero (f x) i)} + λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → rUnitₖ zero (f x) i)} rUnitₕ∙ (suc zero) = ST.elim (λ _ → isOfHLevelPath 2 § _ _) λ {(f , p) → cong ∣_∣₂ (ΣPathP ((λ i x → rUnitₖ 1 (f x) i) , λ i j → rUnitₖ 1 (p j) i))} @@ -556,7 +556,7 @@ lUnitₕ∙ : {A : Pointed ℓ} (n : ℕ) (x : coHomRed n A) → 0ₕ∙ n +[ n ]ₕ∙ x ≡ x lUnitₕ∙ zero = ST.elim (λ _ → isOfHLevelPath 2 § _ _) - λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → lUnitₖ zero (f x) i)} + λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → lUnitₖ zero (f x) i)} lUnitₕ∙ (suc zero) = ST.elim (λ _ → isOfHLevelPath 2 § _ _) λ {(f , p) → cong ∣_∣₂ (ΣPathP ((λ i x → lUnitₖ 1 (f x) i) , λ i j → lUnitₖ 1 (p j) i))} @@ -580,7 +580,7 @@ rCancelₕ∙ : {A : Pointed ℓ} (n : ℕ) (x : coHomRed n A) → x +[ n ]ₕ∙ (-[ n ]ₕ∙ x) ≡ 0ₕ∙ n rCancelₕ∙ zero = ST.elim (λ _ → isOfHLevelPath 2 § _ _) - λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → rCancelₖ zero (f x) i)} + λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → rCancelₖ zero (f x) i)} rCancelₕ∙ {A = A} (suc zero) = ST.elim (λ _ → isOfHLevelPath 2 § _ _) λ {(f , p) → cong ∣_∣₂ (ΣPathP ((λ i x → rCancelₖ 1 (f x) i) , λ i j → rCancelₖ 1 (p j) i))} @@ -593,7 +593,7 @@ lCancelₕ∙ : {A : Pointed ℓ} (n : ℕ) (x : coHomRed n A) → (-[ n ]ₕ∙ x) +[ n ]ₕ∙ x ≡ 0ₕ∙ n lCancelₕ∙ zero = ST.elim (λ _ → isOfHLevelPath 2 § _ _) - λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → lCancelₖ zero (f x) i)} + λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) λ i x → lCancelₖ zero (f x) i)} lCancelₕ∙ {A = A} (suc zero) = ST.elim (λ _ → isOfHLevelPath 2 § _ _) λ {(f , p) @@ -610,7 +610,7 @@ assocₕ∙ zero = ST.elim3 (λ _ _ _ → isOfHLevelPath 2 § _ _) λ {(f , p) (g , q) (h , r) - → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) + → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) (λ i x → assocₖ zero (f x) (g x) (h x) i))} assocₕ∙ (suc zero) = ST.elim3 (λ _ _ _ → isOfHLevelPath 2 § _ _) diff --git a/Cubical.ZCohomology.Groups.CP2.html b/Cubical.ZCohomology.Groups.CP2.html index d5bc678301..9f010bed3c 100644 --- a/Cubical.ZCohomology.Groups.CP2.html +++ b/Cubical.ZCohomology.Groups.CP2.html @@ -286,7 +286,7 @@ compEquiv (m.TotalSpaceHopfPush→TotalSpace , m.isEquivTotalSpaceHopfPush→TotalSpace) - (Σ-cong-equiv (idEquiv _) + (Σ-cong-equiv (idEquiv _) λ x → F x , F-eq x) CP²-iso : Iso CP2 (Pushout {A = TotalHopf} (λ _ → tt) fst) diff --git a/Cubical.ZCohomology.Groups.KleinBottle.html b/Cubical.ZCohomology.Groups.KleinBottle.html index 981e9eb624..51895fa5f8 100644 --- a/Cubical.ZCohomology.Groups.KleinBottle.html +++ b/Cubical.ZCohomology.Groups.KleinBottle.html @@ -171,7 +171,7 @@ leftInv Iso-H¹-𝕂²₁ (x , (p , (q , P))) = ΣPathP (refl , (ΣPathP (sym (nilpotent→≡refl x p P) - , toPathP (Σ≡Prop (λ _ → isOfHLevelTrunc 3 _ _ _ _) + , toPathP (Σ≡Prop (λ _ → isOfHLevelTrunc 3 _ _ _ _) (transportRefl q))))) {- But this is precisely the type (minus set-truncation) of H¹(S¹) -} @@ -186,8 +186,8 @@ setTruncIso ( compIso (characFunSpace𝕂² (coHomK 1)) (compIso - (Σ-cong-iso-snd (λ x → Σ-cong-iso-snd - λ p → Σ-cong-iso-snd + (Σ-cong-iso-snd (λ x → Σ-cong-iso-snd + λ p → Σ-cong-iso-snd λ q → movePathIso p q (isCommΩK-based 1 x))) (compIso Iso-H¹-𝕂²₁ Iso-H¹-𝕂²₂))) @@ -238,7 +238,7 @@ {- Step two : ∥ Σ[ p ∈ x ≡ x ] p ∙ p ≡ refl ∥₂ ≡ ∥ Σ[ x ∈ K₁ ] x + x ≡ 0 ∥₂ -} Iso-H²-𝕂²₂ : Iso ∥ (Σ[ p ∈ 0ₖ 2 ≡ 0ₖ 2 ] p ∙ p ≡ refl) ∥₂ ∥ Σ[ x ∈ coHomK 1 ] x +ₖ x ≡ 0ₖ 1 ∥₂ -Iso-H²-𝕂²₂ = setTruncIso (Σ-cong-iso {B' = λ x → x +ₖ x ≡ 0ₖ 1} (invIso (Iso-Kn-ΩKn+1 1)) +Iso-H²-𝕂²₂ = setTruncIso (Σ-cong-iso {B' = λ x → x +ₖ x ≡ 0ₖ 1} (invIso (Iso-Kn-ΩKn+1 1)) λ p → compIso (congIso (invIso (Iso-Kn-ΩKn+1 1))) (pathToIso λ i → ΩKn+1→Kn-hom 1 p p i ≡ 0ₖ 1)) @@ -405,9 +405,9 @@ theIso = compIso (setTruncIso (compIso (characFunSpace𝕂² (coHomK 2)) - (Σ-cong-iso-snd - λ x → Σ-cong-iso-snd - λ p → Σ-cong-iso-snd + (Σ-cong-iso-snd + λ x → Σ-cong-iso-snd + λ p → Σ-cong-iso-snd λ q → (movePathIso p q (isCommΩK-based 2 x))))) (compIso Iso-H²-𝕂²₁ (compIso diff --git a/Cubical.ZCohomology.Groups.Prelims.html b/Cubical.ZCohomology.Groups.Prelims.html index 0bd6959d9b..1f1e57e6c7 100644 --- a/Cubical.ZCohomology.Groups.Prelims.html +++ b/Cubical.ZCohomology.Groups.Prelims.html @@ -138,7 +138,7 @@ {- Proof that (S¹ → K₁) ≃ K₁ × ℤ. Needed for H¹(T²) -} S1→K₁≡S1×ℤ : Iso ((S₊ 1) → coHomK 1) (coHomK 1 × ℤ) -S1→K₁≡S1×ℤ = S¹→S¹≡S¹×ℤ ⋄ prodIso (invIso (truncIdempotentIso 3 (isGroupoidS¹))) idIso +S1→K₁≡S1×ℤ = S¹→S¹≡S¹×ℤ ⋄ prodIso (invIso (truncIdempotentIso 3 (isGroupoidS¹))) idIso module _ (key : Unit') where module P = lockedCohom key diff --git a/Cubical.ZCohomology.Groups.RP2.html b/Cubical.ZCohomology.Groups.RP2.html index 3e3412cda6..3cededafcb 100644 --- a/Cubical.ZCohomology.Groups.RP2.html +++ b/Cubical.ZCohomology.Groups.RP2.html @@ -92,7 +92,7 @@ (T.elim (λ _ → isGroupoidΠ λ _ → isOfHLevelPlus {n = 1} 2 (isSetSetTrunc _ _)) (toPropElim (λ _ → isPropΠ (λ _ → isSetSetTrunc _ _)) λ {(p , nilp) - → cong ∣_∣₂ (ΣPathP (refl , Σ≡Prop (λ _ → isOfHLevelTrunc 3 _ _ _ _) + → cong ∣_∣₂ (ΣPathP (refl , Σ≡Prop (λ _ → isOfHLevelTrunc 3 _ _ _ _) (rUnit refl ∙∙ cong (Kn→ΩKn+1 0) (sym (nilpotent→≡0 (ΩKn+1→Kn 0 p) (sym (ΩKn+1→Kn-hom 0 p p) @@ -104,7 +104,7 @@ contrGroupIsoUnit (isOfHLevelRetractFromIso 0 (setTruncIso (compIso funSpaceIso-RP² - (Σ-cong-iso-snd (λ _ → Σ-cong-iso-snd λ _ → pathIso)))) + (Σ-cong-iso-snd (λ _ → Σ-cong-iso-snd λ _ → pathIso)))) isContr-H¹-RP²-helper) --- H²(RP²) ≅ ℤ/2ℤ ---- @@ -127,7 +127,7 @@ λ p → refl))) Iso-H²-RP²₂ : Iso ∥ Σ[ p ∈ 0ₖ 2 ≡ 0ₖ 2 ] p ≡ sym p ∥₂ Bool -Iso-H²-RP²₂ = compIso (setTruncIso (Σ-cong-iso-snd λ _ → pathIso)) +Iso-H²-RP²₂ = compIso (setTruncIso (Σ-cong-iso-snd λ _ → pathIso)) (compIso Iso-H²-𝕂²₂ testIso) diff --git a/Cubical.ZCohomology.Groups.Torus.html b/Cubical.ZCohomology.Groups.Torus.html index 8d6a131d49..d6933a233b 100644 --- a/Cubical.ZCohomology.Groups.Torus.html +++ b/Cubical.ZCohomology.Groups.Torus.html @@ -163,7 +163,7 @@ H¹-T²≅ℤ×ℤ = theIso □ GroupIsoDirProd (Hⁿ-Sⁿ≅ℤ 0) (H⁰-Sⁿ≅ℤ 0) where typIso : Iso _ _ - typIso = setTruncIso (curryIso ⋄ codomainIso S1→K₁≡S1×ℤ ⋄ toProdIso) + typIso = setTruncIso (curryIso ⋄ codomainIso S1→K₁≡S1×ℤ ⋄ toProdIso) ⋄ setTruncOfProdIso theIso : GroupIso _ _ @@ -213,7 +213,7 @@ , refl) rightInv helper _ = refl theIso : Iso (coHom 2 (S¹ × S¹)) (coHom 1 S¹) - theIso = setTruncIso (curryIso ⋄ codomainIso S1→K2≡K2×K1 ⋄ toProdIso) + theIso = setTruncIso (curryIso ⋄ codomainIso S1→K2≡K2×K1 ⋄ toProdIso) ⋄ setTruncOfProdIso ⋄ helper diff --git a/Cubical.ZCohomology.Groups.Wedge.html b/Cubical.ZCohomology.Groups.Wedge.html index 6ce96318e3..154cd6429b 100644 --- a/Cubical.ZCohomology.Groups.Wedge.html +++ b/Cubical.ZCohomology.Groups.Wedge.html @@ -227,11 +227,11 @@ rightInv (fst H⁰Red-⋁) = uncurry (ST.elim2 (λ _ _ → isOfHLevelPath 2 (isSet× isSetSetTrunc isSetSetTrunc) _ _) - λ {(_ , _) (_ , _) → ΣPathP (cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl) - , cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl))}) + λ {(_ , _) (_ , _) → ΣPathP (cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl) + , cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl))}) leftInv (fst H⁰Red-⋁) = ST.elim (λ _ → isOfHLevelPath 2 isSetSetTrunc _ _) - λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) + λ {(f , p) → cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) (funExt λ {(inl a) → refl ; (inr b) → refl ; (push tt i) j → (cong (p ∙_) (symDistr (cong f (sym (push tt))) p) @@ -242,8 +242,8 @@ snd H⁰Red-⋁ = makeIsGroupHom (ST.elim2 (λ _ _ → isOfHLevelPath 2 (isSet× isSetSetTrunc isSetSetTrunc) _ _) - λ {(f , p) (g , q) → ΣPathP (cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl) - , cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl))}) + λ {(f , p) (g , q) → ΣPathP (cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl) + , cong ∣_∣₂ (Σ≡Prop (λ _ → isSetℤ _ _) refl))}) wedgeConnected : ((x : typ A) → ∥ pt A ≡ x ∥₁) → ((x : typ B) → ∥ pt B ≡ x ∥₁) → (x : A ⋁ B) → ∥ inl (pt A) ≡ x ∥₁ wedgeConnected conA conB = diff --git a/Cubical.ZCohomology.Gysin.html b/Cubical.ZCohomology.Gysin.html index 4ce808e12b..96109d76d9 100644 --- a/Cubical.ZCohomology.Gysin.html +++ b/Cubical.ZCohomology.Gysin.html @@ -1110,7 +1110,7 @@ -- at an exact sequence involving it. ϕ∘j≡ : (i : ℕ) → ϕ∘j i ≡ ⌣-hom i ϕ∘j≡ i = - Σ≡Prop (λ _ → isPropIsGroupHom _ _) + Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (ST.elim (λ _ → isOfHLevelPath 2 squash₂ _ _) λ _ → refl)) diff --git a/Cubical.ZCohomology.Properties.html b/Cubical.ZCohomology.Properties.html index 818b187c33..9be44ce5fd 100644 --- a/Cubical.ZCohomology.Properties.html +++ b/Cubical.ZCohomology.Properties.html @@ -547,7 +547,7 @@ isContr-↓∙ : (n : ℕ) → isContr (coHomK-ptd (suc n) →∙ coHomK-ptd n) fst (isContr-↓∙ zero) = (λ _ → 0) , refl snd (isContr-↓∙ zero) (f , p) = - Σ≡Prop (λ f → isSetℤ _ _) + Σ≡Prop (λ f → isSetℤ _ _) (funExt (T.elim (λ _ → isOfHLevelPath 3 (isOfHLevelSuc 2 isSetℤ) _ _) (toPropElim (λ _ → isSetℤ _ _) (sym p)))) fst (isContr-↓∙ (suc n)) = (λ _ → 0ₖ _) , refl @@ -561,7 +561,7 @@ isContr-↓∙' : (n : ℕ) → isContr (S₊∙ (suc n) →∙ coHomK-ptd n) fst (isContr-↓∙' zero) = (λ _ → 0) , refl snd (isContr-↓∙' zero) (f , p) = - Σ≡Prop (λ f → isSetℤ _ _) + Σ≡Prop (λ f → isSetℤ _ _) (funExt (toPropElim (λ _ → isSetℤ _ _) (sym p))) fst (isContr-↓∙' (suc n)) = (λ _ → 0ₖ _) , refl fst (snd (isContr-↓∙' (suc n)) f i) x = diff --git a/Cubical.ZCohomology.RingStructure.CohomologyRing.html b/Cubical.ZCohomology.RingStructure.CohomologyRing.html index 78029e293f..368751a776 100644 --- a/Cubical.ZCohomology.RingStructure.CohomologyRing.html +++ b/Cubical.ZCohomology.RingStructure.CohomologyRing.html @@ -54,9 +54,9 @@ _⌣_ (λ {k} {l} → 0ₕ-⌣ k l) (λ {k} {l} → ⌣-0ₕ k l) - (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _))))) - (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) - (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) + (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _))))) + (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) + (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) (λ _ _ _ → leftDistr-⌣ _ _ _ _ _) λ _ _ _ → rightDistr-⌣ _ _ _ _ _ diff --git a/Cubical.ZCohomology.RingStructure.CohomologyRingFun.html b/Cubical.ZCohomology.RingStructure.CohomologyRingFun.html index fedc1936bb..e7d93b5d63 100644 --- a/Cubical.ZCohomology.RingStructure.CohomologyRingFun.html +++ b/Cubical.ZCohomology.RingStructure.CohomologyRingFun.html @@ -46,9 +46,9 @@ _⌣_ (λ {k} {l} → 0ₕ-⌣ k l) (λ {k} {l} → ⌣-0ₕ k l) - (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _))))) - (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) - (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) + (λ _ _ _ → sym (ΣPathTransport→PathΣ _ _ ((sym (+'-assoc _ _ _)) , (sym (assoc-⌣ _ _ _ _ _ _))))) + (λ _ → sym (ΣPathTransport→PathΣ _ _ (sym (+'-rid _) , sym (lUnit⌣ _ _)))) + (λ _ → ΣPathTransport→PathΣ _ _ (refl , transportRefl _ ∙ rUnit⌣ _ _)) (λ _ _ _ → leftDistr-⌣ _ _ _ _ _) λ _ _ _ → rightDistr-⌣ _ _ _ _ _