Hostname: page-component-784d4fb959-8cb88 Total loading time: 0 Render date: 2025-07-16T20:33:27.584Z Has data issue: false hasContentIssue false
  • English
  • Français

A parametricity-based formalization of semi-simplicial and semi-cubical sets

Published online by Cambridge University Press:  23 June 2025

Hugo Herbelin Open the ORCID record for Hugo Herbelin [Opens in a new window]  and
Ramkumar Ramachandra Open the ORCID record for Ramkumar Ramachandra [Opens in a new window]
Hugo Herbelin*
Affiliation:
Université Paris Cité, Inria, CNRS, IRIF, Paris, France
Ramkumar Ramachandra
Affiliation:
Université Paris Cité (2020-2022), Unaffiliated, Paris, France
*
Corresponding author: Hugo Herbelin; Email: Hugo.Herbelin@inria.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Semi-simplicial and semi-cubical sets are commonly defined as presheaves over, respectively, the semi-simplex or semi-cube category. Homotopy type theory then popularized an alternative definition, where the set of $n$-simplices or $n$-cubes are instead regrouped into the families of the fibers over their faces, leading to a characterization we call indexed. Moreover, it is known that semi-simplicial and semi-cubical sets are related to iterated Reynolds parametricity, respectively, in their unary and binary variants. We exploit this correspondence to develop an original uniform indexed definition of both augmented semi-simplicial and semi-cubical sets, and fully formalize it in Coq.

Keywords

semi-simplicial setssemi-cubical setsparametricityCoqformalization

Information

Type
Special Issue: Advances in Homotopy type theory
Information
Mathematical Structures in Computer Science , Volume 35 , 2025 , e13
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Altenkirch, T., Capriotti, P. and Kraus, N. (2016). Extending homotopy type theory with strict equality. In: Talbot, J. and Regnier, L. (eds.) 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29 - September 1, 2016, Marseille, France, LIPIcs, vol. 62, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 21:1–21:17.Google Scholar
Altenkirch, T. and Kaposi, A. (2015). Towards a cubical type theory without an interval. In: Uustalu, T. (ed.) 21st International Conference on Types for Proofs and Programs, TYPES 2015, May 18-21, 2015, Tallinn, Estonia, LIPIcs, vol. 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 3:1–3:27.Google Scholar
Angiuli, C., Brunerie, G., Coquand, T., Harper, R. , Hou (Favonia), K. B. and Licata, D. R. (2021). Syntax and models of cartesian cubical type theory. Mathematical Structures in Computer Science 31 (4) 424–468.10.1017/S0960129521000347CrossRefGoogle Scholar
Annenkov, D., Capriotti, P. and Kraus, N. (2017). Two-level type theory and applications. CoRR, abs/1705.03307.Google Scholar
Annenkov, D., Capriotti, P., Kraus, N. and Sattler, C. (2023). Two-level type theory and applications. Mathematical Structures in Computer Science 33 (8) 688–743.10.1017/S0960129523000130CrossRefGoogle Scholar
Atkey, R., Ghani, N. and Johann, P. (2014). A relationally parametric model of dependent type theory. In: The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL’14, San Diego, CA, USA, January 20-21, 2014.Google Scholar
Bernardy, J., Coquand, T. and Moulin, G. (2015). A presheaf model of parametric type theory. Electronic Notes in Theoretical Computer Science (319) 67–82.10.1016/j.entcs.2015.12.006CrossRefGoogle Scholar
Bernardy, J., Jansson, P. and Paterson, R. (2010). Parametricity and dependent types. In: Hudak, P. and Weirich, S.(eds.) Proceeding of the 15th ACM SIGPLAN International Conference on Functional Programming, ICFP 2010, September 27-29, 2010, Baltimore, Maryland, USA, ACM, 345–356.Google Scholar
Bernardy, J. and Lasson, M. (2011). Realizability and parametricity in pure type systems. In Foundations of Software Science and Computational Structures - 14th International Conference, FOSSACS. 2011, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS. 2011. Proceedings, Saarbrücken, Germany, March 26-April 3, 2011, Lecture Notes in Computer Science, vol. 6604, Springer, 108–122.Google Scholar
Bernardy, J.-P. and Moulin, G. (2012). A computational interpretation of parametricity. In: 2012 27th Annual IEEE Symposium on Logic in Computer Science, IEEE, 135–144.10.1109/LICS.2012.25CrossRefGoogle Scholar
Bezem, M., Coquand, T. and Huber, S. (2013a). 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, April 22-26, 2013, Toulouse, France, LIPIcs, vol. 26, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 107–128.Google Scholar
Bezem, M., Coquand, T. and Huber, S. (2013b). A model of type theory in cubical sets. In: Matthes, R. and Schubert, A.(eds.) 19th International Conference oon Types for Proofs and Programs, TYPES 2013, April 22-26, 2013, Toulouse, France, LIPIcs, vol. 26. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 107–128.Google Scholar
Buchholtz, U. and Morehouse, E. (2017). Varieties of cubical sets. In: International Conference on Relational and Algebraic Methods in Computer Science, Springer, 77–92.Google Scholar
Cavallo, E. and Harper, R. (2020). Internal parametricity for cubical type theory. In: Fernández, M. and Muscholl, A. (eds.) 28th EACSL Annual Conference on Computer Science Logic (CSL 2020), Dagstuhl, Germany, Leibniz International Proceedings in Informatics (LIPIcs), vol. 152, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 13:1–13:17.Google Scholar
Cohen, C., Coquand, T., Huber, S. and Mörtberg, A. (2015). Cubical type theory: a constructive interpretation of the univalence axiom. In: Uustalu, T. (ed.) 21st International Conference oon Types for Proofs and Programs, TYPES 2015, May 18-21, 2015, Tallinn, Estonia, LIPIcs, vol. 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 5:1–5:34.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.) TYPES 2015, LIPIcs, vol. 69, Schloss Dagstuhl, 5:1–5:34.Google Scholar
Curien, P.-L., Garner, R. and Hofmann, M. (2014). Revisiting the categorical interpretation of dependent type theory. Theoretical Computer Science 546 99–119. Models of Interaction: Essays in Honour of Glynn Winskel.10.1016/j.tcs.2014.03.003CrossRefGoogle Scholar
Gilbert, G., Cockx, J., Sozeau, M. and Tabareau, N. (2019). Definitional proof-irrelevance without K. Proceedings of the ACM on Programming Languages 3 (POPL) 3:1–3:28.10.1145/3290316CrossRefGoogle Scholar
Grandis, M. and Mauri, L. (2003). Cubical sets and their site. Theory and Applications of Categories 11 185–211.Google Scholar
Herbelin, H. (2015). A dependently-typed construction of semi-simplicial types. Mathematical Structures in Computer Science 25 (5) 1116–1131.10.1017/S0960129514000528CrossRefGoogle Scholar
Herbelin, H. (2020). Investigations into syntactic iterated parametricity and cubical type theory. Available at https://hott-uf.github.io/2020/HoTTUF_2020_paper_22.pdf.Google Scholar
Hofmann, M. (1995). Extensional concepts in intensional type theory. PhD thesis, University of Edinburgh.Google Scholar
Hofmann, M. and Streicher, T. (1994). The groupoid model refutes uniqueness of identity proofs. In: Proceedings of the Ninth Annual Symposium on Logic in Computer Science (LICS’94), Paris, France, July 4-7, 1994, IEEE Computer Society, 208–212.10.1109/LICS.1994.316071CrossRefGoogle Scholar
Johann, P. and Sojakova, K. (2017). Cubical categories for higher-dimensional parametricity. CoRR, abs/1701.06244.Google Scholar
Kapulkin, K. and Lumsdaine, P. L. (2021). The simplicial model of univalent foundations (after Voevodsky). Journal of the European Mathematical Society 23 (6) 2071–2126.10.4171/jems/1050CrossRefGoogle Scholar
Kraus, N. (2021). Internal $\infty$ -categorical models of dependent type theory: towards 2LTT eating HoTT. In: 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29–July 2, 2021, IEEE, 1–14.Google Scholar
Kraus, N. and Sattler, C. (2017). Space-valued diagrams, type-theoretically. arXiv preprint arXiv:1704.04543.Google Scholar
Lasson, M. (2014). Réalisabilité et Paramétricité dans les Systèmes de Types Purs. PhD thesis, Université de Lyon.Google Scholar
Loregian, F. and Riehl, E. (2020). Categorical notions of fibration. Expositiones Mathematicae 38 (4) 496–514.10.1016/j.exmath.2019.02.004CrossRefGoogle Scholar
Martin-Löf, P. (1975). An intuitionistic theory of types, predicative part. In Logic Colloquium, North Holland, 73–118.Google Scholar
Martin-Löf, P. (1984). Intuitionistic Type Theory, Studies in Proof Theory, vol. 1, Bibliopolis.Google Scholar
Moeneclaey, H. (2021). Parametricity and semi-cubical types. In: 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29 - July 2, 2021, IEEE, 1–11.Google Scholar
Moeneclaey, H. (2022). Cubical models are cofreely parametric. PhD thesis, Université Paris Cité.Google Scholar
Moulin, G. (2016). Internalizing parametricity. PhD thesis, Department of Computer Science and Engineering, Chalmers University of Technology.Google Scholar
Nuyts, A. and Devriese, D. (2024). Transpension: the right adjoint to the i-type. Logical Methods in Computer Science 20 (2).Google Scholar
Nuyts, A., Vezzosi, A. and Devriese, D. (2017). Parametric quantifiers for dependent type theory. Proceedings of the ACM on Programming Languages 1 (ICFP) 1–29.10.1145/3110276CrossRefGoogle Scholar
Part, F. and Luo, Z. (2015). Semi-simplicial types in logic-enriched homotopy type theory. CoRR, abs/1506.04998.Google Scholar
Reynolds, J. C. (1983). Types, abstraction and parametric polymorphism. In: Mason, R. E. A. (ed.) Information Processing 83, Proceedings of the IFIP 9th World Computer Congress, Paris, France, September 19-23, 1983, North-Holland/IFIP, 513–523.Google Scholar
Shulman, M. (2015). Univalence for inverse diagrams and homotopy canonicity. Mathematical Structures in Computer Science 25 (5) 1203–1277.10.1017/S0960129514000565CrossRefGoogle Scholar
The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study..Google Scholar
Voevodsky, V. (2012). Semi-simplicial types. Briefly described in Section 8 of Herbelin (2015).Google Scholar