The Univalence Axiom in Cubical Sets

Simon Huber,Marc Bezem,Thierry Coquand

doi:10.1007/s10817-018-9472-6

Simon Huber, Marc Bezem + Show 1 more

Open Access

https://doi.org/10.1007/s10817-018-9472-6

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Abstract

In this note we show that Voevodsky’s univalence axiom holds in the model of type theory based on cubical sets as described in Bezem et al. (in: Matthes and Schubert (eds.) 19th international conference on types for proofs and programs (TYPES 2013), Leibniz international proceedings in informatics (LIPIcs), Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, vol 26, pp 107–128, 2014. https://doi.org/10.4230/LIPIcs.TYPES.2013.107. http://drops.dagstuhl.de/opus/volltexte/2014/4628) and Huber (A model of type theory in cubical sets. Licentiate thesis, University of Gothenburg, 2015). We will also discuss Swan’s construction of the identity type in this variation of cubical sets. This proves that we have a model of type theory supporting dependent products, dependent sums, univalent universes, and identity types with the usual judgmental equality, and this model is formulated in a constructive metatheory.

Highlights

We give a brief overview of the cubical set model, introducing some different notations, but will otherwise assume the reader is familiar with [2,6]
As opposed to [2,6] let us define cubical sets as contravariant presheaves on the opposite of the category used there, that is, the category of cubes C contains as objects finite sets I = {i1, . . . , in} (n ≥ 0) of names and a morphism f : J → I is given by a set-theoretic map I → J ∪ {0, 1} which is injective when restricted to the preimage of J ; we will write compositions in applicative order
In [2] we showed that Kan types are closed under dependent products and sums constituting a model of type theory

Summary

Review of the Cubical Set Model

We give a brief overview of the cubical set model, introducing some different notations, but will otherwise assume the reader is familiar with [2,6]. If Γ is a cubical set, we write Ty(Γ ) for the collection/class of presheaves on the category of elements of Γ [2,6]. Such a presheaf A ∈ Ty(Γ ) is given by a family of sets A(I, ρ) for I ∈ C and ρ ∈ Γ (I ) together with restriction functions. Any Kan structure κ defines a composition operation κwhich provides the missing lid of the open box, given by: κρ [J → u; (i, 0) → ui0] = (κ ρ [J → u; (i, 0) → ui0])(i/1). In [2] we showed that Kan types are closed under dependent products and sums constituting a model of type theory

Path Types

Equivalences and Univalence

Identity Types