eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-03-15
5:1
5:34
10.4230/LIPIcs.TYPES.2015.5
article
Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom
Cohen, Cyril
Coquand, Thierry
Huber, Simon
Mörtberg, Anders
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol069-types2015/LIPIcs.TYPES.2015.5/LIPIcs.TYPES.2015.5.pdf
univalence axiom
dependent type theory
cubical sets