Skip to main content Accessibility help
×
Home
Hostname: page-component-846f6c7c4f-msmtk Total loading time: 0.267 Render date: 2022-07-06T14:37:04.948Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue

Constructive sheaf models of type theory

Published online by Cambridge University Press:  18 November 2021

Thierry Coquand*
Affiliation:
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
Fabian Ruch
Affiliation:
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
Christian Sattler
Affiliation:
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
*
*Corresponding author. Email: Thierry.Coquand@cse.gu.se
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.

Type
Paper
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, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Aczel, P. (1998). On relating type theories and set theories. In: Altenkirch, T., Naraschewski, W. and Reus, B. (eds.) Types for Proofs and Programs, International Workshop TYPES’98, Kloster Irsee, Germany, March 27–31, 1998, Selected Papers, Lecture Notes in Computer Science, vol. 1657, Springer, 1–18.Google Scholar
Angiuli, C., Brunerie, G., Coquand, T., Hou (Favonia), K.-B., Harper, R. and Licata, D. R. (2017). Cartesian cubical type theory. Draft.Google Scholar
Avigad, J., Kapulkin, K. and Lumsdaine, P. L. (2015). Homotopy limits in type theory. Mathematical Structures in Computer Science 25 (5) 10401070.10.1017/S0960129514000498CrossRefGoogle Scholar
Barr, M. (1974). Toposes without points. Journal of Pure and Applied Algebra 5 265280.10.1016/0022-4049(74)90037-1CrossRefGoogle Scholar
Bergner, J. E. and Rezk, C. (2013). Reedy categories and the Θ-construction. Mathematische Zeitschrift 274 (1–2) 499514.10.1007/s00209-012-1082-0CrossRefGoogle Scholar
Beth, E. W. (1956). Semantic Construction of Intuitionistic Logic. Medededlingen der koninklijke Nederlandse Akademie van Wetenschappen, afd. Letterkunde. Nieuwe Reeks, Deel 19, No. 11. N. V. Noord-Hollandsche Uitgevers Maatschappij, Amsterdam.Google Scholar
Boulier, S. P. (2018). Extending Type Theory with Syntactic Models. (Etendre la théorie des types à l’aide de modèles syntaxiques). Phd thesis, Ecole nationale supérieure Mines-Télécom Atlantique Bretagne Pays de la Loire, France.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 on Types for Proofs and Programs, TYPES 2015, May 18–21, 2015, Tallinn, Estonia, LIPIcs, vol. 69, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 5:1–5:34.Google Scholar
Coquand, T., Huber, S. and Mörtberg, A. (2018). On higher inductive types in cubical type theory. In: Dawar, A. and Grädel, E. (eds.) Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09–12, 2018, ACM, 255264.10.1145/3209108.3209197CrossRefGoogle Scholar
Coquand, T., Mannaa, B. and Ruch, F. (2017). Stack semantics of type theory. In: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20–23, 2017, IEEE Computer Society, 1–11.10.1109/LICS.2017.8005130CrossRefGoogle Scholar
Coquand, T. and Paulin, C. (1988). Inductively defined types. In: Martin-Löf, P. and Mints, G. (eds.) COLOG-88, International Conference on Computer Logic, Tallinn, USSR, December 1988, Proceedings, Lecture Notes in Computer Science, vol. 417, Springer, 5066.Google Scholar
Dybjer, P. (1995). Internal type theory. In: Berardi, S. and Coppo, M. (eds.) Types for Proofs and Programs, International Workshop TYPES’95, Torino, Italy, June 5–8, 1995, Selected Papers, Lecture Notes in Computer Science, vol. 1158, Springer, 120–134.Google Scholar
Eilenberg, S. and Zilber, J. A. (1950). Semi-simplicial complexes and singular homology. Annals of Mathematics (2) 51 499513.10.2307/1969364CrossRefGoogle Scholar
Grothendieck, A. (1960). Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4 228.10.1007/BF02684778CrossRefGoogle Scholar
Hofmann, M. (1997). Syntax and semantics of dependent types. In: Semantics and Logics of Computation (Cambridge, 1995), Publications of the Newton Institute, vol. 14, Cambridge, Cambridge University Press, 79130.10.1017/CBO9780511526619.004CrossRefGoogle Scholar
Joyal, A. (1984). Lettre à Grothendieck.Google Scholar
Kaposi, A., Huber, S. and Sattler, C. (2019). Gluing for type theory. In: Geuvers, H. (ed.) 4th International Conference on Formal Structures for Computation and Deduction, FSCD 2019, June 24–30, 2019, Dortmund, Germany, LIPIcs, vol. 131, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 25:1–25:19.Google Scholar
Kraus, N. (2015). Truncation Levels in Homotopy Type Theory. Phd thesis, University of Nottingham, UK.Google Scholar
Kripke, S. A. (1965). Semantical analysis of intuitionistic logic. I. In: Formal Systems and Recursive Functions (Proc. Eighth Logic Colloq., Oxford, 1963), North-Holland, Amsterdam, 92130.10.1016/S0049-237X(08)71685-9CrossRefGoogle Scholar
Mannaa, B. and Coquand, T. (2013). Dynamic Newton-Puiseux theorem. Journal of Logic & Analysis 5.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, August 29 - September 1, 2016, Marseille, France, LIPIcs, vol. 62, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 24:1–24:19.Google Scholar
Quirin, K. (2016). Lawvere-Tierney Sheafification in Homotopy Type Theory. (Faisceautisation de Lawvere-Tierney en théorie des types homotopiques). Phd thesis, École des mines de Nantes, France.Google Scholar
Rezk, C. (2010). Toposes and homotopy toposes. Unpublished manuscript.Google Scholar
Rijke, E., Shulman, M. and Spitters, B. (2020). Modalities in homotopy type theory. Logical Methods in Computer Science 16 (1).Google Scholar
Sattler, C. (2017). The equivalence extension property and model structures. CoRR, abs/1704.06911.Google Scholar
Schreiber, U. and Shulman, M. (2012). Quantum gauge field theory in cohesive homotopy type theory. In: Duncan, R. and Panangaden, P. (eds.) Proceedings 9th Workshop on Quantum Physics and Logic, QPL 2012, Brussels, Belgium, 10–12 October 2012, EPTCS, vol. 158, 109126.Google Scholar
Scott, D. S. (1980). Relating theories of the λ-calculus. In: To HB. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, London-New York, Academic Press, 403450.Google Scholar
Shulman, M. (2015a). The univalence axiom for elegant Reedy presheaves. Homology, Homotopy and Applications 17 (2) 81106.10.4310/HHA.2015.v17.n2.a6CrossRefGoogle Scholar
Shulman, M. (2015b). Univalence for inverse diagrams and homotopy canonicity. Mathematical Structures in Computer Science 25 (5) 12031277.10.1017/S0960129514000565CrossRefGoogle Scholar
Shulman, M. (2018). Brouwer’s fixed-point theorem in real-cohesive homotopy type theory. Mathematical Structures in Computer Science 28 (6) 856941.10.1017/S0960129517000147CrossRefGoogle Scholar
Shulman, M. (2019). All (∞ ,1)-toposes have strict univalent universes. CoRR, abs/1904.07004.Google Scholar
Swan, A. and Uemura, T. (2019). On Church’s thesis in cubical assemblies. CoRR, abs/1905.03014.Google Scholar
Troelstra, A. S. and van Dalen, D. (1988). Constructivism in mathematics. Vol. II, Studies in Logic and the Foundations of Mathematics, vol. 123, Amsterdam, North-Holland Publishing Co. An introduction.Google Scholar
Univalent Foundations Program, T. (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study.Google Scholar
Voevodsky, V. (2015). An experimental library of formalized mathematics based on the univalent foundations. Mathematical Structures in Computer Science 25 (5) 12781294.10.1017/S0960129514000577CrossRefGoogle Scholar
Weaver, M. Z. and Licata, D. R. (2020). A constructive model of directed univalence in bicubical sets. In: Hermanns, H., Zhang, L., Kobayashi, N. and Miller, D. (eds.) LICS’20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8–11, 2020, ACM, 915928.Google Scholar
Wellen, F. (2017). Formalizing Cartan Geometry in Modal Homotopy Type Theory. Phd thesis, Karlsruher Institut für Technologie, Germany.Google Scholar
You have Access Open access
1
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Constructive sheaf models of type theory
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Constructive sheaf models of type theory
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Constructive sheaf models of type theory
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *