Niels van der Weide, The internal languages of univalent categories

Homotopy Type Theory Electronic Seminar Talks, 2024-11-21
https://www.uwo.ca/math/faculty/kapul...

Internal language theorems are fundamental in categorical logic, since they express an equivalence between syntax and semantics. One of such theorems was proven by Clairambault and Dybjer, who corrected the result originally by Seely. More specifically, they constructed a biequivalence between the bicategory of locally Cartesian closed categories and the bicategory of democratic categories with families with extensional identity types, ∑-types, and ∏-types. This theorem expresses that, up to adjoint equivalence, the internal language of locally Cartesian closed categories is extensional Martin-Löf type theory with dependent sums and products.

In this talk, we prove the theorem by Clairambault and Dybjer for univalent categories, and we extend their biequivalence to various classes of toposes, among which are ∏-pretoposes and elementary toposes. Univalent categories give an interesting framework for studying internal language theorems of dependent type theory. This is because of the fact that univalent categories are identified up to adjoint equivalence, and that internal language theorems give us biequivalence for various classes of categories and theories. In addition, we shall see that univalence gives us several ways to simplify the necessary constructions and proofs, because it allows us to transfer properties and structure along equivalences for free.

The results in this paper have been formalized using the proof assistant Coq and the UniMath library. The material in this talk is based on the preprint https://arxiv.org/abs/2411.06636.