Path Categories and Propositional Identity Types

Abstract

Connections between homotopy theory and type theory have recently attracted a lot of attention, with Voevodsky’s univalent foundations and the interpretation of Martin-Löf’s identity types in Quillen model categories as some of the highlights. In this article, we establish a connection between a natural weakening of Martin-Löf’s rules for the identity types that has been considered by Cohen, Coquand, Huber and Mörtberg in their work on a constructive interpretation of the univalence axiom on the one hand and the notion of a path category, a slight variation on the classic notion of a category of fibrant objects due to Brown, on the other. This involves showing that the syntactic category associated to a type theory with weak identity types carries the structure of a path category, strengthening earlier results by Avigad, Lumsdaine, and Kapulkin. In this way, we not only relate a well-known concept in homotopy theory with a natural concept in logic but also provide a framework for further developments.