Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.14/173481
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- Title
- Topological and simplicial models of identity types
- Related
- ACM transactions on computational logic, Vol. 13, No. 1, (2012), p.3-1-3-44
- DOI
- 10.1145/2071368.2071371
- Publisher
- Association for Computing Machinery
- Date
- 2012
- Author/Creator
- Van Den Berg, Benno
- Author/Creator
- Garner, Richard
- Description
- 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 [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 Top and 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 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.
- Description
- 44 page(s)
- Subject Keyword
- theory
- Subject Keyword
- dependent type theory
- Subject Keyword
- identity types
- Subject Keyword
- categorical semantics
- Subject Keyword
- homotopy theory
- Resource Type
- journal article
- Organisation
- Macquarie University. Dept. of Computing
- Identifier
- http://hdl.handle.net/1959.14/173481
- Identifier
- ISSN:1529-3785
- Identifier
- mq-rm-2011006942
- Identifier
- mq_res-ext-2-s2.0-84857170576
- Language
- eng
- Reviewed
