The simplicial model of Univalent Foundations (after Voevodsky)

Peter LeFanu Lumsdaine,Krzysztof Kapulkin

doi:10.4171/jems/1050

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-Löf type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

Highlights

The Univalent Foundations programme is a new proposed approach to foundations of mathematics, originally suggested by Vladimir Voevodsky in [Voe06], building on the systems of dependent type theory developed by Martin-Lof and others
Equality carries no information beyond its truth-value: if two things are equal, they are equal in at most one way. This is fine for equality between elements of discrete sets; but it is unnatural for objects of categories, or points of spaces. It is at odds with the informal mathematical practice of treating isomorphic objects as equal; which is why this usage must be so often disclaimed as an abuse of language, and kept rigorously away from formal statements, even though it is so appealing
We focus on the Quillen model category sSet of simplicial sets, a well-studied model for topological spaces in homotopy theory; we construct a model of type theory in sSet, and show that it satisfies the Univalence Axiom

Summary

Introduction

The Univalent Foundations programme is a new proposed approach to foundations of mathematics, originally suggested by Vladimir Voevodsky in [Voe06], building on the systems of dependent type theory developed by Martin-Lof and others. The main goal of this paper is to justify the intuition outlined above, of types as spaces To this end, we focus on the Quillen model category sSet of simplicial sets, a well-studied model for topological spaces in homotopy theory; we construct a model of type theory in sSet, and show that it satisfies the Univalence Axiom. In particular the closed types will be interpreted as Kan complexes, which serve as a model for ∞-groupoids, for instance in Joyal and Lurie’s approach to higher category theory It follows from this model that Martin-Lof type theory plus the Univalence Axiom (presented in terms of contextual categories) is consistent, provided that the classical foundations we use are—precisely, ZFC together with the existence of two strongly inaccessible cardinals, or equivalently two Grothendieck universes. Univalence in homotopy-theoretic settings is considered in [GK17]. (These references are, far from exhaustive.)

Models from Universes

The Simplicial Model

Findings

Univalence