License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TYPES.2015.5
URN: urn:nbn:de:0030-drops-84754
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8475/
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Cohen, Cyril ; Coquand, Thierry ; Huber, Simon ; Mörtberg, Anders

Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

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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.

BibTeX - Entry

@InProceedings{cohen_et_al:LIPIcs:2018:8475,
  author =	{Cyril Cohen and Thierry Coquand and Simon Huber and Anders M{\"o}rtberg},
  title =	{{Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom}},
  booktitle =	{21st International Conference on Types for Proofs and Programs (TYPES 2015)},
  pages =	{5:1--5:34},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-030-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{69},
  editor =	{Tarmo Uustalu},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8475},
  URN =		{urn:nbn:de:0030-drops-84754},
  doi =		{10.4230/LIPIcs.TYPES.2015.5},
  annote =	{Keywords: univalence axiom, dependent type theory, cubical sets}
}

Keywords: univalence axiom, dependent type theory, cubical sets
Collection: 21st International Conference on Types for Proofs and Programs (TYPES 2015)
Issue Date: 2018
Date of publication: 15.03.2018

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