### Abstract

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.

Original language | English |
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Article number | 3 |

Pages (from-to) | 3:1-3:44 |

Number of pages | 44 |

Journal | ACM Transactions on Computational Logic |

Volume | 13 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2012 |

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## Cite this

*ACM Transactions on Computational Logic*,

*13*(1), 3:1-3:44. [3]. https://doi.org/10.1145/2071368.2071371