NDLI: Topological and Simplicial Models of Identity Types

Content Provider	ACM Digital Library
Author	Van den berg, Benno Garner, Richard
Copyright Year	2012
Abstract	In this paper we construct new categorical models for the identity types of Martin-Löf type theory, in the categories $\textbf{Top}$ of topological spaces and $\textbf{SSet}$ of simplicial sets. We do so building on earlier work of Awodey and Warren [2009], which has suggested that a suitable environment for the interpretation of identity types should be a category equipped with a weak factorization 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 $\textbf{Top}$ and $\textbf{SSet}$ are categories endowed with a weak factorization 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 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 $\textbf{Top}$ and $\textbf{SSet},$ 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.
Starting Page	1
Ending Page	44
Page Count	44
File Format	PDF
ISSN	15293785
e-ISSN	1557945X
DOI	10.1145/2071368.2071371
Volume Number	13
Issue Number	1
Journal	ACM Transactions on Computational Logic (TOCL)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2012-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Dependent type theory Categorical semantics Homotopy theory Identity types
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Logic Computational Mathematics Computer Science

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