2010/07/27 by Benno van den Berg, Berg, Benno van den, Richard Garner +1
Computer Science · Mathematics · #03B15 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Primary: 18C50 #Secondary: 55U35
paper · pdf · doi:10.48550/arxiv.1007.4638
openalex publication_date 2010/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we construct new categorical models for the identity types of Martin-Löf 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.