The simplicial model of Univalent Foundations (after Voevodsky)

In this largely expository paper, we construct and investigate a model of the Univalent Foundations of Mathematics in the category of simplicial sets. To this end, we first give a new technique for constructing 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 this model. As a corollary, we conclude that Univalent Foundations are at least as consistent as ZFC with two inaccessible cardinals. The main results of the paper are due to Vladimir Voevodsky.

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