The internal languages of univalent categories

TL;DR

This paper extends the internal language theorem for locally Cartesian closed categories to univalent categories and various topos classes, formalizing results with proof assistants.

Contribution

It generalizes the internal language theorem to univalent categories and toposes, providing formalized proofs using Rocq and UniMath.

Findings

Extended the internal language theorem to univalent categories.

Applied the theorem to various classes of toposes.

Formalized the results in proof assistants Rocq and UniMath.

Abstract

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, $\sum$-types, and $\prod$-types. This theorem expresses that the internal language of locally Cartesian closed categories is extensional Martin-L\"of type theory with dependent sums and products. In this paper, we study the theorem by Clairambault and Dybjer for univalent categories, and we extend it to various classes of toposes, among which are $\prod$-pretoposes, elementary toposes, and elementary toposes with a universe. The results in this paper have been formalized using the proof assistant Rocq and the UniMath library.