一句话：将内部语言定理推广到单值范畴与多种拓扑斯，并用 Rocq/UniMath 形式化。
内部语言定理表达语法与语义之间的等价。Clairambault 与 Dybjer 修正 Seely 的结果，建立了局部笛卡尔闭范畴与外延 Martin-Löf 类型论（含 identity、Σ、Π 类型）的民主 categories with families 之间的双范畴等价。本文把该定理推广到 univalent 范畴，并进一步扩展到 ∏-pretoposes、初等拓扑斯等拓扑斯类。作者在证明助手 Rocq 及其 UniMath 库中形式化了所有结果，从而为这些范畴的内部语言给出机器验证的语义等价。
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, Σ-types, and ∏-types. This theorem expresses that the internal language of locally Cartesian closed categories is extensional Martin-Löf 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 ∏-pretoposes and elementary toposes. The results in this paper have been formalized using the proof assistant Rocq and the UniMath library.
DOI 原文 ·
@inproceedings{paperbot8657,
title = {The internal languages of univalent categories},
author = {Niels van der Weide},
booktitle = {LICS},
year = {2025},
doi = {10.1109/lics65433.2025.00016}
}