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Results tagged with homotopy-type-theory
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user 22131
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
Assuming decidable equality but not LEM in HoTT
Decidability of every set implies the law of excluded middle as soon as there is a "subobject classifier".
Indeed, for every proposition $U$, the fact that "$U = \mathsf{True}$ or $U \neq \mathsf{True …
- 42.4k
7
votes
Accepted
Are lists in homotopy type theory free $A_\infty$-spaces?
In an informal sense, the answer "should be yes", in the sense that if one ignore type theory and work with an $\infty$-topos one can make sense of the construction $List(A)$ either by the usual unive …
- 42.4k
9
votes
What kind of category is generated by Cubical type theory?
I would say that the question is not even well defined.
Saying that Martin löf type theory with extensional identity types is the internal language of cartesian closed categories with natural number …
- 42.4k
21
votes
2
answers
1k
views
$\infty$-categorical interpretation of type theory
One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity …
- 42.4k
73
votes
Accepted
Why did Voevodsky consider categories "posets in the next dimension", and groupoids the corr...
First, there is indeed nothing mathematically very deep in this observation, and I agree that the word "breakthrough" might be exaggerated. But on the other hand lots of very deep ideas look trivial o …
- 42.4k