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Results tagged with homotopy-type-theory
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user 483446
The homotopy interpretation of constructive dependent type theory, the univalence axiom, higher inductive types, internal languages of higher toposes, univalent foundations for mathematics, and implementations of such theories in proof assistants.
6
votes
1
answer
151
views
Univalence for weakly Tarski universes
In Martin-Löf type theory, a weakly Tarski universe is a type $\mathcal{U}$ with a type family $\mathcal{T}(A)$ indexed by terms $A:\mathcal{U}$, which is closed under identity types, dependent produc …
- 2,624
0
votes
0
answers
90
views
Univalence and higher inductive types in the lambda calculus model of type theory
In appendix A1 of the homotopy type theory book by the Univalent Foundations Project, the authors give a formal presentation of Martin-Löf type theory in lambda calculus. However, they did not give an …
- 2,624
2
votes
1
answer
545
views
Higher inductive types in higher observational type theory
Mike Shulman gave the following set of talks on higher observational type theory earlier this year (part 1, part 2, part 3). However, while he talked about how the identity types are defined and behav …
- 2,624
2
votes
1
answer
462
views
Path types and identity types in dependent type theory
There's been some debate at the nLab recently over the names of "identity type" and "path type" in certain dependent type theories.
One user wrote that
Many cubical type theorists make the distinctio …
- 2,624
1
vote
0
answers
76
views
Directly proving the extensionality principle for product types without quasi-inverses
In section 2.6 of the Univalent Foundations Project's Homotopy Type Theory book, the extensionality principle of product types is proven by showing that for all elements $a:A$, $a':A$, $b:A$, $b':A$, …
- 2,624
1
vote
1
answer
73
views
Constructing set-truncations of types from universes
This is a follow-up question from my previous question titled Constructing coproduct types and boolean types from universes, where I showed how every basic operation in set theory/topos theory could b …
- 2,624
1
vote
Accepted
Constructing set-truncations of types from universes
Given a universe $U$, the type of $U$-small propositions is given by $$\mathrm{Prop} \equiv \sum_{P:U} \prod_{x:P} \prod_{y:P} x = y$$
Given a type $A:U$, for $x:A$ and $y:A$, the type
$$[x = y] \equi …
- 2,624
4
votes
1
answer
186
views
Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory
In Mike Shulman's article Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, the fundamental axiom adopted for his real-cohesive homotopy type theory (axiom $\mathbb{R}\flat$), which …
- 2,624
3
votes
0
answers
277
views
Principle of unique choice in homotopy type theory
In the MathOverflow thread Mathematics without the principle of unique choice, Mike Shulman defines the principle of unique choice to be
if $R$ is a relation between two sets $A$, $B$, and for every …
- 2,624
3
votes
0
answers
74
views
Discreteness of the higher inductive-inductive Cauchy real numbers in real cohesive homotopy...
We work in cohesive homotopy type theory with propositional resizing, so that there is only one type of Dedekind real numbers $\mathbb{R}$ up to equivalence, and Mike Shulman's axiom $\mathbb{R}\flat$ …
- 2,624
5
votes
1
answer
169
views
Are lists in homotopy type theory free $A_\infty$-spaces?
Traditionally in dependent type theory with axiom K or uniqueness of identity proofs, every type $A$ is 0-truncated, and thus the type of lists on $A$, $\mathrm{List}(A)$, is 0-truncated and the free …
- 2,624
5
votes
Consistency of Generalised Continuum Hypothesis and univalence in HoTT
Dominik Kirst and Felix Rech managed to formalize the Generalized Continuum Hypothesis for sets in the HoTT library in Coq with univalence, which already shows the consistency of the GCH with univalen …
- 2,624