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Unifying Cubical Models of Univalent Type Theory

Authors Evan Cavallo , Anders Mörtberg, Andrew W Swan

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Author Details

Evan Cavallo
  • School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Anders Mörtberg
  • School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
  • Department of Mathematics, Stockholm University, Sweden
Andrew W Swan
  • Institute for Logic, Language and Computation, University of Amsterdam, The Netherlands


We are grateful to Carlo Angiuli, Thierry Coquand, Kuen-Bang Hou (Favonia), Robert Harper, Daniel R. Licata, Andrew Pitts and Jon Sterling for helpful comments and remarks on earlier versions of this work. The first two authors are also thankful to Mathieu Anel and Steve Awodey for their illuminating lectures on homotopical algebra in the CMU HoTT seminar.

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Evan Cavallo, Anders Mörtberg, and Andrew W Swan. Unifying Cubical Models of Univalent Type Theory. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 14:1-14:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • Theory of computation → Type theory
  • Cubical Set Models
  • Cubical Type Theory
  • Homotopy Type Theory
  • Univalent Foundations


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