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Syntax and models of Cartesian cubical type theory

Published online by Cambridge University Press:  17 January 2022

Carlo Angiuli Open the ORCID record for Carlo Angiuli [Opens in a new window] ,
Guillaume Brunerie ,
Thierry Coquand Open the ORCID record for Thierry Coquand [Opens in a new window] ,
Robert Harper ,
Kuen-Bang Hou (Favonia) Open the ORCID record for Kuen-Bang Hou (Favonia) [Opens in a new window]  and
Daniel R. Licata Open the ORCID record for Daniel R. Licata [Opens in a new window]
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.

Keywords

Type theoryhomotopy type theorycubical type theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 4 , April 2021 , pp. 424 - 468
Creative Commons
Creative Common License - CCCreative Common License - BY
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

References

Angiuli, C. (2019). Computational Semantics of Cartesian Cubical Type Theory. Phd thesis, Carnegie Mellon University. Technical Report CMU-CS-19-127. Available from http://reports-archive.adm.cs.cmu.edu/anon/2019/abstracts/19-127.html.Google Scholar
Angiuli, C., Cavallo, E., Hou (Favonia), K.-B., Harper, R. and Sterling, J. (2018a). The RedPRL proof assistant (invited paper). In: Blanqui, F. and Reis, G. (eds.) 13th International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice (LFMTP 2018), Electronic Proceedings in Theoretical Computer Science, vol. 274.CrossRefGoogle Scholar
Angiuli, C. and Harper, R. (2017). Computational higher type theory II: dependent cubical realizability. Preprint. https://arxiv.org/abs/1606.09638.Google Scholar
Angiuli, C., Harper, R. and Wilson, T. (2016). Computational higher type theory I: abstract cubical realizability. Preprint. http://arxiv.org/abs/1604.08873.Google Scholar
Angiuli, C., Harper, R. and Wilson, T. (2017a). Computational higher-dimensional type theory. In: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages, POPL 2017, New York, NY, USA. ACM, 680–693.CrossRefGoogle Scholar
Angiuli, C., Hou (Favonia), K.-B. and Harper, R. (2017b). Computational higher type theory III: univalent universes and exact equality. Preprint. https://arxiv.org/abs/1712.01800.CrossRefGoogle Scholar
Angiuli, C., Hou (Favonia), K.-B. and Harper, R. (2018b). Cartesian cubical computational type theory: constructive reasoning with paths and equalities. In: Computer Science Logic.Google Scholar
Awodey, S. (2016). Notes on cubical sets. Available from https://github.com/awodey/math/blob/master/Cubical/cubical.pdf.Google Scholar
Awodey, S. (2018a). An algebraic fibrant replacement for cubical sets. Talk at Category Theory OctoberFest.Google Scholar
Awodey, S. (2018b). A cubical model of homotopy type theory. Annals of Pure and Applied Logic 169 (12) 1270–1294. Logic Colloquium 2015.CrossRefGoogle Scholar
Bauer, A., EscardÓ, M., Lumsdaine, P. L. and Mahboubi, A. (2019). Formalization of mathematics in type theory (Dagstuhl Seminar 18341). Dagstuhl Reports 8 (8) 130–155.Google Scholar
Bentzen, B. (2019). Naive cubical type theory. Preprint. https://arxiv.org/abs/1911.05844.Google Scholar
Bernardy, J.-P., Coquand, T. and Moulin, G. (2015). A presheaf model of parametric type theory. In: Mathematical Foundations of Programming Semantics.Google Scholar
Bezem, M. and Coquand, T. (2015). A Kripke model for simplicial sets. Theoretical Computer Science 574 86–91.CrossRefGoogle Scholar
Bezem, M., Coquand, T. and Huber, S. (2014). A model of type theory in cubical sets. In: Matthes, R. and Schubert, A. (eds.) 19th International Conference on Types for Proofs and Programs (TYPES 2013), 107–128.Google Scholar
Bezem, M., Coquand, T. and Huber, S. (2019). The univalence axiom in cubical sets. Journal of Automated Reasoning 63 (2) 159–171.10.1007/s10817-018-9472-6CrossRefGoogle Scholar
Bickford, M. (2020). Formalization of cubical type theory in Nuprl. http://nuprl.org/KB/show.php?ID=774.Google Scholar
Birkedal, L., Bizjak, A., Clouston, R., Grathwohl, H. B., Spitters, B. and Vezzosi, A. (2019). Guarded cubical type theory. Journal of Automated Reasoning 63 (2) 211–253.10.1007/s10817-018-9471-7CrossRefGoogle Scholar
Brunerie, G. and Licata, D. R. (2014). A cubical infinite-dimensional type theory. Talk at Oxford Workshop on Homotopy Type Theory.Google Scholar
Buchholtz, U. and Morehouse, E. (2017). Varieties of cubical sets. In: Relational and Algebraic Methods in Computer Science: 16th International Conference, RAMiCS 2017, Lyon, France, May 15–18, 2017, Proceedings. Springer International Publishing.Google Scholar
Cavallo, E. (2021). Higher Inductive Types and Internal Parametricity for Cubical Type Theory. Phd thesis, Carnegie Mellon University. Technical Report CMU-CS-21-100. Available from http://reports-archive.adm.cs.cmu.edu/anon/2021/CMU-CS-21-100.pdf.Google Scholar
Cavallo, E. and Harper, R. (2019). Higher inductive types in cubical computational type theory. Proceedings of the ACM on Programming Languages 3 (POPL) 1:1–1:27.Google Scholar
Cavallo, E., MÖrtberg, A. and Swan, A. W. (2020). Unifying cubical models of univalent type theory. In: FernÁndez, M. and Muscholl, A. (eds.) 28th EACSL Annual Conference on Computer Science Logic (CSL 2020), Leibniz International Proceedings in Informatics (LIPIcs), vol. 152, Dagstuhl, Germany. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 14:1–14:17.Google Scholar
Cisinski, D.-C. (2006). Les prÉfaisceaux comme modÈles des types d’homotopie. AstÉrisque, Soc. Math. France 308.Google Scholar
Cohen, C., Coquand, T., Huber, S. and MÖrtberg, A. (2018). Cubical type theory: a constructive interpretation of the univalence axiom. In: Uustalu, T. (ed.) 21st International Conference on Types for Proofs and Programs (TYPES 2015), 5:1–5:34.Google Scholar
Constable, R. L., Allen, S. F., Bromley, H. M., Cleaveland, W. R., Cremer, J. F., Harper, R. W., Howe, D. J., Knoblock, T. B., Mendler, N. P., Panangaden, P., Sasaki, J. T. and Smith, S. F. (1986). Implementing Mathematics with the NuPRL Proof Development System. Prentice Hall.Google Scholar
Coquand, T. (2014a). Variations on cubical sets. Talk at Oxford Workshop on Homotopy Type Theory. Available from http://www.cse.chalmers.se/coquand/comp.pdf.Google Scholar
Coquand, T. (2014b). Variations on cubical sets (diagonals version). Available from http://www.cse.chalmers.se/coquand/diag.pdf.Google Scholar
Coquand, T. (2015). Higher inductive types as cubical sets. Available from http://www.cse.chalmers.se/coquand/hit1.pdf.Google Scholar
Coquand, T., Huber, S. and MÖrtberg, A. (2018). On higher inductive types in cubical type theory. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, New York, NY, USA. ACM, 255–264.CrossRefGoogle Scholar
Coquand, T., Huber, S. and Sattler, C. (2019a). Homotopy canonicity for cubical type theory. In: Geuvers, H. (ed.) 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019), Leibniz International Proceedings in Informatics (LIPIcs), vol. 131, Dagstuhl, Germany. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 11:1–11:23.Google Scholar
Coquand, T., Ruch, F. and Sattler, C. (2019b). Constructive sheaf models of type theory. Preprint. https://arxiv.org/abs/1912.10407.Google Scholar
Gambino, N. and Sattler, C. (2017). The Frobenius condition, right properness, and uniform fibrations. Journal of Pure and Applied Algebra 221 (12) 3027–3068.CrossRefGoogle Scholar
Harper, R. (2018). Computational type theory. Lectures at Oregon Programming Languages Summer School 2018.Google Scholar
Harper, R. and Angiuli, C. (2018). Computational (higher) type theory. Tutorial at POPL 2018.Google Scholar
Hofmann, M. (1995). Extensional Concepts in Intensional Type Theory. Phd thesis, University of Edinburgh.Google Scholar
Hofmann, M. and Streicher, T. (1997). Lifting Grothendieck universes. Available from http://www.mathematik.tu-darmstadt.de/streicher/NOTES/lift.pdf.Google Scholar
Huber, S. (2019). Canonicity for cubical type theory. Journal of Automated Reasoning 63 (2) 173–210.CrossRefGoogle Scholar
Isaev, V. (2014). Homotopy type theory with an interval type. Available from https://valis.github.io/doc.pdf.Google Scholar
Kan, D. M. (1955). Abstract homotopy. I. Proceedings of the National Academy of Sciences of the United States of America 41 (12) 1092–1096.CrossRefGoogle Scholar
Kapulkin, K. and Lumsdaine, P. L. (2021). The simplicial model of Univalent Foundations (after Voevodsky). Journal of the European Mathematical Society 23 2071–2126.CrossRefGoogle Scholar
Licata, D. (2016). Weak univalence with “beta” implies full univalence. Email to Homotopy Type Theory mailing list. https://groups.google.com/d/msg/homotopytypetheory/j2KBIvDw53s/YTDK4D0NFQAJ.Google Scholar
Licata, D. R., Orton, I., Pitts, A. M. and Spitters, B. (2018). Internal universes in models of homotopy type theory. In: Formal Structures for Computation and Deduction (FSCD). arXiv:1801.07664.Google Scholar
Orton, I. and Pitts, A. M. (2016). Axioms for modelling cubical type theory in a topos. In: Talbot, J.-M. and Regnier, L. (eds.) 25th EACSL Annual Conference on Computer Science Logic (CSL 2016), Leibniz International Proceedings in Informatics (LIPIcs), vol. 62, Dagstuhl, Germany. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 24:1–24:19.Google Scholar
Orton, I. and Pitts, A. M. (2018). Axioms for modelling cubical type theory in a topos (extended version). Logical Methods in Computer Science 14 (4:23) 1–33.Google Scholar
Orton, R. I. (2019a). Agda code accompanying Phd thesis: Cubical Models of Homotopy Type Theory – An Internal Approach. https://doi.org/10.17863/CAM.35681.CrossRefGoogle Scholar
Orton, R. I. (2019b). Cubical Models of Homotopy Type Theory - An Internal Approach. Phd thesis, University of Cambridge. https://doi.org/10.17863/CAM.36690.CrossRefGoogle Scholar
Pitts, A. M. (2015). Nominal presentation of cubical sets models of type theory. In: 20th International Conference on Types for Proofs and Programs (TYPES 2014). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.Google Scholar
Riehl, E. (2019). The equivariant uniform Kan fibration model of cubical homotopy type theory. Talk at the International Conference on Homotopy Type Theory (HoTT 2019).Google Scholar
Riehl, E. and Shulman, M. (2017). A type theory for synthetic Y-categories. Higher Structures 1 (1) 147–224.Google Scholar
Sattler, C. (2017). The equivalence extension property and model structures. Preprint. https://arxiv.org/abs/1704.06911.Google Scholar
Sattler, C. (2018). Do cubical models of type theory also model homotopy types? Talk at Workshop on Types, Homotopy Type theory, and Verification at Hausdorff Institute.Google Scholar
Shulman, M. (2019). All (Y,1)-toposes have strict univalent universes. Preprint. https://arxiv.org/abs/1904.07004.Google Scholar
Sterling, J. and Angiuli, C. (2021). Normalization for cubical type theory. In: 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’21, 1–15.CrossRefGoogle Scholar
Sterling, J., Angiuli, C. and Gratzer, D. (2019). Cubical syntax for reflection-free extensional equality. In: Geuvers, H. (ed.) 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019), Leibniz International Proceedings in Informatics (LIPIcs), vol. 131, Dagstuhl, Germany. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 31:1–31:25.Google Scholar
Swan, A. (2016). An algebraic weak factorization system on 01-substitution sets: a constructive proof. Journal of Logic & Analysis 8 (1) 1–35.Google Scholar
The Univalent Foundations Program, Institute for Advanced Study (2013). Homotopy Type Theory: Univalent Foundations Of Mathematics. Available from homotopytypetheory.org/book.Google Scholar
Vezzosi, A., MÖrtberg, A. and Abel, A. (2019). Cubical Agda: a dependently typed programming language with univalence and higher inductive types. Proceedings of the ACM on Programming Languages 3 (ICFP) 87:1–87:29.Google Scholar
Voevodsky, V. (2006). A very short note on homotopy l-calculus. Unpublished. http://www.math.ias.edu/vladimir/files/2006_09_Hlambda.pdf.Google Scholar
Weaver, M. Z. and Licata, D. R. (2020). A constructive model of directed univalence in bicubical sets. In: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’20, New York, NY, USA. Association for Computing Machinery, 915–928.CrossRefGoogle Scholar