Axioms for Modelling Cubical Type Theory in a Topos

TL;DR

This paper investigates the minimal axioms needed within a topos to model cubical type theory, focusing on paths as functions and the properties of uniform Kan filling related to univalence.

Contribution

It identifies weak axioms for modeling path-based type theory in a topos, clarifying the notion of uniform Kan filling and its role in interpreting univalence.

Findings

Established weak axioms for path modeling in a topos

Clarified the properties of uniform Kan filling

Linked axioms to the constructive interpretation of univalence

Abstract

The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an object $I$ in a topos to give such a path-based model of type theory in which paths are just functions with domain $I$. Cohen, Coquand, Huber and M\"ortberg give such a model using a particular category of presheaves. We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose. This clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of their constructive interpretation of Voevodsky's univalence axiom. (This paper is a revised and expanded version of a paper of the same name that appeared in the proceedings of the 25th EACSL Annual Conference on Computer Science Logic, CSL 2016.)