12
votes
2answers
643 views

What is the status of (universal) algebra in type theory?

With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...
6
votes
1answer
212 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 ...
8
votes
2answers
770 views

A (very naive) question about Homotopy Type Theory

In homotopy type theory, homotopy types can be viewed as logical types and it is possible to prove some theorems about them without using any underlying space (no simplicial set, no topological ...
13
votes
2answers
516 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 ...
16
votes
2answers
721 views

Prospects for reverse mathematics in Homotopy Type Theory

Reverse mathematics, as I mean here, is the study of which theorems/axioms can be used to prove other theorems/axioms over a weak base theory. Examples include Subsystems of Second Order Arithmetic ...
8
votes
2answers
673 views

Equivalent form of the Univalence Axiom

I'm reading the new HoTT book and I'm wondering about a potential equivalent form of the Univalence Axiom: $(A \simeq B) \simeq (A = B)$. For simplicity, I'm tacitly working in a fixed universe. It ...
10
votes
1answer
1k views

Forcing in Homotopy Type Theory

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...
5
votes
1answer
877 views

categorifying induction in homotopy type theory

In trying to understand homotopy type theory, I stumbled upon the following silly question, which is likely to be trivial for the experts. Let $B=\sqcup_{n\in\Bbb N} BS_n$, which I'd like to think of ...