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Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

Authors Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg

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Cyril Cohen
Thierry Coquand
Simon Huber
Anders Mörtberg

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Cyril Cohen, Thierry Coquand, Simon Huber, and Anders Mörtberg. Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom. In 21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 5:1-5:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.TYPES.2015.5

Abstract

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.

Subject Classification

Keywords
  • univalence axiom
  • dependent type theory
  • cubical sets

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References

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