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| {-# OPTIONS --safe #-} | ||
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| {- | ||
| This file contians a proof that the smash product turns the universe | ||
| of pointed types into a symmetric monoidal wild category. The pentagon | ||
| and hexagon are proved in separate files due to the length of the | ||
| proofs. The remaining identities and the main result are proved here. | ||
| -} | ||
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| module Cubical.HITs.SmashProduct.SymmetricMonoidalCat where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Pointed | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Foundations.GroupoidLaws | ||
| open import Cubical.Foundations.Path | ||
| open import Cubical.Foundations.Equiv | ||
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| open import Cubical.Data.Unit | ||
| open import Cubical.Data.Sigma using (ΣPathP) | ||
| open import Cubical.Data.Bool | ||
| open import Cubical.HITs.SmashProduct.Base | ||
| open import Cubical.HITs.SmashProduct.Pentagon | ||
| open import Cubical.HITs.SmashProduct.Hexagon | ||
| open import Cubical.HITs.SmashProduct.SymmetricMonoidal | ||
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| open import Cubical.WildCat.Base | ||
| open import Cubical.WildCat.Functor | ||
| open import Cubical.WildCat.Product | ||
| open import Cubical.WildCat.BraidedSymmetricMonoidal | ||
| open import Cubical.WildCat.Instances.Pointed | ||
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| open WildCat | ||
| open WildFunctor | ||
| open isSymmetricWildCat | ||
| open isMonoidalWildCat | ||
| open WildNatIso | ||
| open WildNatTrans | ||
| open wildIsIso | ||
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| private | ||
| variable | ||
| ℓ ℓ' : Level | ||
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| -- ⋀ as a functor | ||
| ⋀F : ∀ {ℓ} → WildFunctor (PointedCat ℓ × PointedCat ℓ) (PointedCat ℓ) | ||
| F-ob ⋀F (A , B) = A ⋀∙ B | ||
| F-hom ⋀F (f , g) = f ⋀→∙ g | ||
| F-id ⋀F = ⋀→∙-idfun | ||
| F-seq ⋀F (f , g) (f' , g') = ⋀→∙-comp f f' g g' | ||
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| ⋀lUnitNatIso : WildNatIso (PointedCat ℓ) (PointedCat ℓ) | ||
| (restrFunctorₗ ⋀F Bool*∙) (idWildFunctor (PointedCat ℓ)) | ||
| N-ob (trans ⋀lUnitNatIso) X = ≃∙map ⋀lIdEquiv∙ | ||
| N-hom (trans ⋀lUnitNatIso) f = ⋀lId-sq f | ||
| inv' (isIs ⋀lUnitNatIso c) = ≃∙map (invEquiv∙ ⋀lIdEquiv∙) | ||
| sect (isIs (⋀lUnitNatIso {ℓ = ℓ}) c) = | ||
| ≃∙→ret/sec∙ (⋀lIdEquiv∙ {ℓ = ℓ} {A = c}) .snd | ||
| retr (isIs ⋀lUnitNatIso c) = | ||
| ≃∙→ret/sec∙ ⋀lIdEquiv∙ .fst | ||
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| makeIsIso-Pointed : ∀ {ℓ} {A B : Pointed ℓ} {f : A →∙ B} | ||
| → isEquiv (fst f) → wildIsIso {C = PointedCat ℓ} f | ||
| inv' (makeIsIso-Pointed {f = f} eq) = ≃∙map (invEquiv∙ ((fst f , eq) , snd f)) | ||
| sect (makeIsIso-Pointed {f = f} eq) = ≃∙→ret/sec∙ ((fst f , eq) , snd f) .snd | ||
| retr (makeIsIso-Pointed {f = f} eq) = ≃∙→ret/sec∙ ((fst f , eq) , snd f) .fst | ||
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| restrₗᵣ : WildNatIso (PointedCat ℓ) (PointedCat ℓ) | ||
| (restrFunctorᵣ ⋀F Bool*∙) (restrFunctorₗ ⋀F Bool*∙) | ||
| N-ob (trans restrₗᵣ) X = ⋀comm→∙ | ||
| N-hom (trans restrₗᵣ) f = ⋀comm-sq f (idfun∙ Bool*∙) | ||
| isIs restrₗᵣ c = makeIsIso-Pointed (isoToIsEquiv ⋀CommIso) | ||
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| -- main result | ||
| ⋀Symm : ∀ {ℓ} → isSymmetricWildCat (PointedCat ℓ) | ||
| _⊗_ (isMonoidal ⋀Symm) = ⋀F | ||
| 𝟙 (isMonoidal ⋀Symm) = Bool*∙ | ||
| N-ob (trans (⊗assoc (isMonoidal ⋀Symm))) (A , B , C) = ≃∙map SmashAssocEquiv∙ | ||
| N-hom (trans (⊗assoc (isMonoidal ⋀Symm))) (f , g , h) = ⋀assoc-⋀→∙ f g h | ||
| inv' (isIs (⊗assoc (isMonoidal ⋀Symm)) (A , B , C)) = | ||
| ≃∙map (invEquiv∙ SmashAssocEquiv∙) | ||
| sect (isIs (⊗assoc (isMonoidal ⋀Symm)) (A , B , C)) = | ||
| ≃∙→ret/sec∙ SmashAssocEquiv∙ .snd | ||
| retr (isIs (⊗assoc (isMonoidal ⋀Symm)) (A , B , C)) = | ||
| ≃∙→ret/sec∙ SmashAssocEquiv∙ .fst | ||
| ⊗lUnit (isMonoidal ⋀Symm) = ⋀lUnitNatIso | ||
| ⊗rUnit (isMonoidal ⋀Symm) = compWildNatIso _ _ _ restrₗᵣ ⋀lUnitNatIso | ||
| triang (isMonoidal (⋀Symm {ℓ})) X Y = ⋀triang | ||
| ⊗pentagon (isMonoidal ⋀Symm) X Y Z W = | ||
| (∘∙-assoc assc₅∙ assc₄∙ assc₃∙) ∙ pentagon∙ | ||
| N-ob (trans (Braid ⋀Symm)) X = ⋀comm→∙ | ||
| N-hom (trans (Braid ⋀Symm)) (f , g) = ⋀comm-sq f g | ||
| isIs (Braid ⋀Symm) _ = makeIsIso-Pointed (isoToIsEquiv ⋀CommIso) | ||
| isSymmetricWildCat.hexagon ⋀Symm a b c = hexagon∙ | ||
| symBraiding ⋀Symm X Y = | ||
| ΣPathP ((funExt (Iso.rightInv ⋀CommIso)) , (sym (rUnit refl))) |
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