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arXiv · 1211.2851

The Simplicial Model of Univalent Foundations (after Voevodsky)

Chris Kapulkin·Peter LeFanu Lumsdaine
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Abstract

We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Martin-L\"of type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

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Chris Kapulkin, Peter LeFanu Lumsdaine. 2012-11-12. The Simplicial Model of Univalent Foundations (after Voevodsky). https://doi.org/10.4171/jems%2F1050

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