Cubical Type Theory: a constructive interpretation of the univalence axiom
Logic in Computer Science
2016-11-14 v1 Logic
Abstract
This paper presents a type theory in which it is possible to directly manipulate -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
Cite
@article{arxiv.1611.02108,
title = {Cubical Type Theory: a constructive interpretation of the univalence axiom},
author = {Cyril Cohen and Thierry Coquand and Simon Huber and Anders Mörtberg},
journal= {arXiv preprint arXiv:1611.02108},
year = {2016}
}
Comments
To be published in the post-proceedings of the 21st International Conference on Types for Proofs and Programs, TYPES 2015