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Results tagged with homotopy-type-theory
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user 11640
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.
4
votes
Accepted
Does the induction principle of the type of booleans imply its recursion principle?
Recursion principles are usually special cases of induction principles in dependent type theory. For example, for the boolean type $\mathbf{2}$, given $a_0, a_1 : A$, we may form (using the induction …
- 15.9k
16
votes
Accepted
Why the reflection rule trivializes higher paths in Martin-Löf Extensional Type theory?
The point is that the reflection rule makes $p = \mathsf{refl}_x$ a well-formed expression. This turns out to be incredibly dangerous: now we can prove it by induction on equality.
More precisely:
…
- 15.9k
10
votes
Accepted
How should I be thinking about object classifiers / universal fibrations / universes?
Yes, the universe is the classifying space for small homotopy types. For various reasons, the universe is not itself a small homotopy type; so it fails to be an object classifier for the trivial reaso …
- 15.9k
8
votes
1
answer
638
views
The independence of path induction
In §1.12 of the Homotopy type theory book, it is mentioned that indiscernibility of identicals is a consequence of path induction. More precisely, for each type $C$ dependent over a type $A$, there is …
- 15.9k
19
votes
3
answers
1k
views
What are finite homotopy types?
Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’:
The homotopy type of a simplicial set that has only finitely many n …
- 15.9k
16
votes
2
answers
674
views
How to formulate the univalence axiom without universes?
The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot …
- 15.9k