Axioms for Modelling Cubical Type Theory in a Topos

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 Mörtberg 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.)

References in corpus (3)

• The Frobenius Condition, Right Properness, and Uniform Fibrations
• Internal Universes in Models of Homotopy Type Theory
• Guarded Cubical Type Theory: Path Equality for Guarded Recursion

Cited by in corpus (9)

• Two-Level Type Theory and Applications
• Towards a constructive simplicial model of Univalent Foundations
• Internal Parametricity for Cubical Type Theory
• Transpension: The Right Adjoint to the Pi-type
• Kripke-Joyal forcing for type theory and uniform fibrations
• From Cubes to Twisted Cubes via Graph Morphisms in Type Theory
• Stable factorization from a fibred algebraic weak factorization system
• Multimodal Dependent Type Theory
• Unifying cubical and multimodal type theory