2026-07-22T04:53:00Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8475 2024-03-06T10:39:27Z ddc:004 open_access

Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom Cohen, Cyril Coquand, Thierry Huber, Simon Mörtberg, Anders univalence axiom dependent type theory cubical sets 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Cyril Cohen and Thierry Coquand and Simon Huber and Anders Mörtberg 2018 Is Part Of LIPIcs, Volume 69, 21st International Conference on Types for Proofs and Programs (TYPES 2015) (2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.TYPES.2015.5 urn:nbn:de:0030-drops-84754 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015.5 eng https://creativecommons.org/licenses/by/3.0/legalcode