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Syntax and models of Cartesian cubical type theory

Published online by Cambridge University Press:  17 January 2022

Carlo Angiuli
Affiliation:
Carnegie Mellon University, Pittsburgh, PA15213, USA
Guillaume Brunerie
Affiliation:
Stockholm University, Stockholm, Sweden
Thierry Coquand
Affiliation:
University of Gothenburg, Gothenburg, Sweden
Robert Harper
Affiliation:
Carnegie Mellon University, Pittsburgh, PA15213, USA
Kuen-Bang Hou (Favonia)
Affiliation:
University of Minnesota, Minneapolis, MN55455, USA
Daniel R. Licata*
Affiliation:
Wesleyan University, Middletown, CT06459, USA
*
*Corresponding author. Email: dlicata@wesleyan.edu
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Abstract

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We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, suspension, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgmental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of generating trivial cofibrations and a different set of generating cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model of this type theory in Cartesian cubical sets. We give a mechanized proof, using Agda as the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, suspension types, and the universe itself are Kan in this model, and that the universe is univalent. An advantage of this formal approach is that our construction can also be interpreted in a range of other models, including cubical sets on the connections cube category and the De Morgan cube category, as used in the CCHM model, and bicubical sets, as used in directed type theory.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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