Topological and simplicial models of identity types

TL;DR

This paper develops new categorical models for identity types in Martin-Löf type theory using topological spaces and simplicial sets, introducing a homotopy-theoretic framework to address coherence issues and construct sound models.

Contribution

It introduces the notion of a path object category and constructs homotopy-theoretic models of identity types in Top and SSet, advancing categorical interpretations of type theory.

Findings

Constructed homotopy-theoretic models in Top and SSet.

Introduced the concept of a path object category.

Provided a framework for sound interpretation of identity types.

Abstract

In this paper we construct new categorical models for the identity types of Martin-L\"of type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do so building on earlier work of Awodey and Warren, which has suggested that a suitable environment for the interpretation of identity types should be a category equipped with a weak factorisation system in the sense of Bousfield--Quillen. It turns out that this is not quite enough for a sound model, due to some subtle coherence issues concerned with stability under substitution; and so our first task is to introduce a slightly richer structure---which we call a homotopy-theoretic model of identity types---and to prove that this is sufficient for a sound interpretation. Now, although both Top and SSet are categories endowed with a weak factorisation system---and indeed, an entire Quillen model structure---exhibiting the additional structure required for a homotopy-theoretic model is quite hard to do. However, the categories we are interested in share a number of common features, and abstracting away from these leads us to introduce the notion of a path object category. This is a relatively simple axiomatic framework, which is nonetheless sufficiently strong to allow the construction of homotopy-theoretic models. Now by exhibiting suitable path object structures on Top and SSet, we endow those categories with the structure of a homotopy-theoretic model: and in this way, obtain the desired topological and simplicial models of identity types.