JournalsjemsVol. 23, No. 6pp. 2071–2126

The simplicial model of Univalent Foundations (after Voevodsky)

  • Krzysztof Kapulkin

    University of Pittsburgh, USA
  • Peter LeFanu Lumsdaine

    Institute for Advanced Study, Princeton, USA
The simplicial model of Univalent Foundations (after Voevodsky) cover
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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-Löf type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

Cite this article

Krzysztof Kapulkin, Peter LeFanu Lumsdaine, The simplicial model of Univalent Foundations (after Voevodsky). J. Eur. Math. Soc. 23 (2021), no. 6, pp. 2071–2126

DOI 10.4171/JEMS/1050