Cubical Type Theory: a constructive interpretation of the univalence axiom

Cyril Cohen (1) , Thierry Coquand (2) , Simon Huber (2) , Anders Mörtberg (3)

1 MARELLE - Mathematical, Reasoning and Software
2 CSE - Department of Computer Science and Engineering [Göteborg]
3 School of Mathematics [Princeton]

This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

Keywords: Univalence Axiom; Dependent Type Theory; Cubical Sets

Domains: Formal Languages and Automata Theory [cs.FL]; Algebraic Topology [math.AT]; Category Theory [math.CT]

Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg. Cubical Type Theory: a constructive interpretation of the univalence axiom. 21st International Conference on Types for Proofs and Programs, May 2015, Tallinn, Estonia. pp.262, ⟨10.4230/LIPIcs.TYPES.2015.5⟩. ⟨hal-01378906v2⟩