-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathindex.pug
198 lines (173 loc) · 11.9 KB
/
index.pug
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
include header
html
head
meta(property='og:title' content='URS')
meta(property='og:description' content='Super Type Theory')
meta(property='og:url' content='https://urs.groupoid.space/')
block title
title URS
block content
nav
<a href='#'>URS</a>
article.main
.exe
section
h1 HOMOTOPY SUPER SMOOTH <br> DIFFERENTIAL QUANTUM <br> TYPE SYSTEM
aside
time Published: 28 FEB 2025
+tex.
We present Homotopy Supersmooth Differential Quantum Type System.
This system builds a progressive structure for formalizing mathematical and physical concepts,
from homotopy and higher categorical structures,
through geometric cohesion and differential properties,
to the rich graded and equivariant world of supergeometry.
Inspired by Urs Schreiber’s work on equivariant super homotopy theory,
this layered approach offers a modular,
type-theoretic foundation for synthetic supergeometry and beyond.
+tex.
Type theory has emerged as a powerful language for mathematics
and physics, unifying computation, logic, and structure. This
article introduces a layered type theory that extends
Martin-Löf’s intensional type theory into a framework capable
of capturing homotopy, cohesion, and supergeometry:
+tex.
1) Homotopy Type System: foundation for homotopical properties of spaces;
2) Super Smooth Type Theory: the graded universes/tensors, group actions, and super-modality operators for geometric cohesion and differential structure such as bosonic/fermionic, flat/sharp modalities;
3) Differential K-Theoretical Type Theory: homotopical foundations via spectra and groupoids, supporting differential K-theory with forms, connections, and refined cohomology structures;
4) Quantum Linear Type Theory: configuration spaces and braids for quantum statistics, supporting quantum systems and linear types for resource-sensitive quantum programming.
+tex.
Each layer builds on the previous, culminating in a system tailored
to formalize superpoints ($\mathbb{R}^{m|n}$), supersymmetry, and equivariant structures,
as exemplified in Schreiber’s "Equivariant Super Homotopy Theory" (2019).
section
.macro
.macro__col
h3#hts <b>HOMOTOPY</b>
ol
li: a(href='https://urs.groupoid.space/foundations/universe/') UNIVERSE</sub>
li: a(href='https://urs.groupoid.space/foundations/forall/') FORALL
li: a(href='https://urs.groupoid.space/foundations/sigma/') SIGMA
li: a(href='https://urs.groupoid.space/foundations/exists/') EXISTS
li: a(href='https://urs.groupoid.space/foundations/real/') REAL
li: a(href='https://urs.groupoid.space/foundations/path/') PATH
li: a(href='https://urs.groupoid.space/foundations/glue/') GLUE
.macro__col
h3#mltt <b>SUP-SMTH</b>
ol
li: a(href='https://urs.groupoid.space/foundations/graded/') GRADED
li: a(href='https://urs.groupoid.space/foundations/flat/') FLAT
li: a(href='https://urs.groupoid.space/foundations/sharp/') SHARP
li: a(href='https://urs.groupoid.space/foundations/bose/') BOSE
li: a(href='https://urs.groupoid.space/foundations/fermi/') FERMI
li: a(href='https://urs.groupoid.space/foundations/tensor/') TENSOR
li: a(href='https://urs.groupoid.space/foundations/smth/') SMTH
li: a(href='https://urs.groupoid.space/foundations/supsmth/') SUPSMTH
.macro__col
h3#mltt <b>DIFF-K</b>
ol
li: a(href='https://urs.groupoid.space/foundations/spectra/') SPECTRA
li: a(href='https://urs.groupoid.space/foundations/grpd/') GRPD
li: a(href='https://urs.groupoid.space/foundations/form/') FORM
li: a(href='https://urs.groupoid.space/foundations/ku/') KU
li: a(href='https://urs.groupoid.space/foundations/diff/') DIFF
li: a(href='https://urs.groupoid.space/foundations/diffku/') DIFFKU
.macro__col
h3#mltt <b>QUANTUM</b>
ol
li: a(href='https://urs.groupoid.space/foundations/conf/') CONF<sup>n</sup>
li: a(href='https://urs.groupoid.space/foundations/braid/') BRAID
li: a(href='https://urs.groupoid.space/foundations/qubit/') QUBIT
li: a(href='https://urs.groupoid.space/foundations/linear/') LINEAR
br
+tex.
$\mathbf{Installation}$.
+tex.
Urs operates as an OCaml framework, manipulating pure ASTs, similar to a Lisp-like
environment, pending a dedicated syntax.
+code.
$ git clone [email protected]:groupoid/urs
$ ocamlfind ocamlc -o urs -package z3 -linkpkg src/urs.ml
section
h1 SYNTAX
+tex(true).
$$
\begin{array}{c} \\
\mathrm{Urs} := \mathrm{MLTT}\ |\ \mathrm{Base}\ |\ \mathrm{Super}\ \\
\mathrm{MLTT} := \mathrm{Cosmos}\ |\ \mathrm{Var}\ |\ \mathrm{Forall}\ |\ \mathrm{Exists}\ \\
|\ \mathrm{\Sigma}\ |\ \mathrm{\Pi}\ |\ \mathbf{Id}\ |\ 0\ |\ 1\ |\ 2\ |\ \mathbf{W} \\
\mathrm{Cosmos} := \mathbf{U_n}\ |\ \mathbf{V_n}\ |\ \mathbf{U_n^{g \in \{Bose,Fermi\}}}\ \\
\mathrm{Var} := \mathbf{var}\ \mathrm{ident}\ |\ \mathbf{hole}\ \\
\mathrm{Base} := \mathbf{ℕ}\ |\ \mathrm{ℤ}\ |\ \mathrm{ℚ}\ |\ \mathrm{ℝ}\ |\ \mathrm{ℂ}\ |\ \mathrm{ℍ}\ |\ \mathrm{𝕆}\ |\ \mathrm{𝕍}\ \\
\mathrm{Super} := \mathrm{Smooth}\ |\ \mathrm{Spectra}\ |\ \mathrm{Grpd}\ \\
|\ \mathrm{Form}\ |\ \mathrm{Diff}\ |\ \mathrm{Quantum}\ \\
\mathrm{Forall} := \forall\ \mathrm{ident}\ \mathrm{E}\ \mathrm{E}\ |\ \lambda\ \mathrm{ident}\ \mathrm{E}\ \mathrm{E}\ |\ \mathrm{E}\ \mathrm{E}\ \\
\mathrm{Exists} := \exists\ \mathrm{ident}\ \mathrm{E}\ \mathrm{E}\ |\ (\mathrm{E}, \mathrm{E})\ |\ \mathrm{E}\mathbf{.1}\ |\ \mathrm{E}\mathbf{.2}\ \\
\mathrm{Smooth} := \mathbf{SmthSet}\ |\ \mathbf{plt}\ |\ \mathbf{\bigcirc}\ |\ \mathbf{\Im}\ |\ \mathbf{\sharp}\ |\ \mathbf{\flat}\ \\
\mathrm{Spectra} := \mathbf{Sp}\ |\ \mathbf{\Sigma}\ |\ \mathbf{\wedge}\ |\ \mathbf{sp}\ \\
\mathrm{Form} := \mathbf{Form}\ |\ \mathbf{f}\ |\ \mathbf{f_\wedge}\ |\ \mathbf{Form_{ind}}\\
\mathrm{Diff} := \mathbf{Diff}\ |\ \mathbf{KU_\flat^G}\ |\ \mathbf{KU^G}\ \\
\mathrm{Grpd} := \mathbf{Grpd}\ |\ \mathbf{\Delta^n}\ |\ \mathbf{Grpd^n_{Ind}}\ \\
\mathrm{Quantum} := \mathbf{Conf^n}\ |\ \mathbf{Braid}\ |\ \mathbf{Linear}\ \\
\mathbf{Qubit}\ |\ \mathbf{appQubit}\ |\ \mathbf{fuseQubit} \\
\\
\end{array}
$$
h1 SEMANTICS
br
h2 Homotopy
+tex.
A foundational framework where types are ∞-groupoids,
and propositions emerge as homotopical structures.
Equality is a space of paths, extensible to higher
dimensions, weaving homotopy theory into type-theoretic reasoning.
With constructs like $\mathbf{U_n}$, $\mathbf{V_n}$, $\mathbf{\Pi}$, $\mathbf{\Sigma}$,
$\mathbf{\exists}$, $\mathbb{R}$, $\mathbf{Path}$, and $\mathbf{Glue}$,
it blends topological invariance with constructive logic.
This system enables synthetic reasoning about continuity,
deformation, and equivalence, treating mathematical spaces
as dynamic objects rich with paths, homotopies, and higher
coherences—a language for capturing the shape of mathematical truth.
h2 Graded Cohesion
+tex.
A cohesive, graded extension of type theory infused with supergeometry,
where types inhabit universes indexed by parity, symmetry, and smoothness.
Using $\mathbf{U_n^g}$, $\mathfrak{s}$, $\mathbf{\flat}$, $\mathbf{\sharp}$, $\mathbf{\bigcirc}$,
$\mathbf{\Im}$, $\mathbf{\otimes}$, $\mathbf{SmthSet}$, and $\mathbf{SupSmthSet}$,
it equips types with tensor structures, group actions, and super-modality
operators to model differential cohesion, infinitesimal geometry,
and graded symmetries. This system unifies the logic of supermanifolds,
derived geometry, and higher smooth stacks, enabling type-theoretic
reasoning about spaces enriched by quantum and differential symmetry.
h2 Differential K-Theory
+tex.
A homotopy-theoretic framework augmented with differential structure,
where types encode spectra, groupoids, and refined cohomological data.
Leveraging $\mathbf{Sp}$, $\Delta^n$, $\mathbf{Grpd}$, $\mathbf{Form}$,
$\mathbf{KU^G}$, $\mathbf{Diff}$, and $\mathbf{KU^G_\flat}$, it internalizes
differential K-theory to provide a synthetic language for characteristic
classes, index theory, and topological invariants. By integrating stable
homotopy, smooth geometry, and categorical semantics, it articulates the
interplay of curvature, bundles, and higher cohomological operations
within a unified type system of paths and forms.
h2 Qunatum Linearity
+tex.
A type-theoretic foundation for quantum systems, internalizing configuration spaces,
braids, and higher categorical symmetries within a resource-sensitive linear logic.
Built on $\mathbf{Conf^n}$, $\mathbf{Braid}$, $\mathbf{Qubit}$, and $\mathbf{Linear}$, it captures quantum statistics via braided
monoidal categories, quantum group actions, and path spaces of indistinguishable particles.
This system supports quantum computation and field theory semantics, using
linear types to model entanglement, interference, and non-classical information flow,
ensuring quantum processes respect spatial and algebraic phase coherence.
section
h1 Bibliography
br
section
h2 TED-K
p.
+tex(true).
<br>
$$
\int
$$
<br><br>
include footer