Skip to main content
×
×
Home

Univalence for inverse diagrams and homotopy canonicity

  • MICHAEL SHULMAN (a1)
Abstract

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky's univalent model in simplicial sets, we obtain new models of univalence in a number of (∞, 1)-toposes; this answers a question raised at the Oberwolfach workshop on homotopical type theory. Second, by gluing the syntactic category of univalent type theory along its global sections functor to groupoids, we obtain a partial answer to Voevodsky's homotopy-canonicity conjecture: in 1-truncated type theory with one univalent universe of sets, any closed term of natural number type is homotopic to a numeral.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@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 sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

      Univalence for inverse diagrams and homotopy canonicity
      Available formats
      ×
      Send article to Dropbox

      To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Univalence for inverse diagrams and homotopy canonicity
      Available formats
      ×
      Send article to Google Drive

      To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Univalence for inverse diagrams and homotopy canonicity
      Available formats
      ×
Copyright
References
Hide All
Arndt, P. and Kapulkin, K. (2011) Homotopy-theoretic models of type theory. In: Ong, L. (ed.) Typed Lambda Calculi and Applications, Lecture Notes in Computer Science volume 6690, Springer Berlin/Heidelberg 4560.
Avigad, J., Kapulkin, C. and Lumsdaine, P. L. (2013) Homotopy limits in Coq. ArXiv:1304.0680.
Awodey, S., Garner, R., Martin-Löf, P. and Voevodsky, V. (2011) Mini-workshop: The homotopical interpretation of constructive type theory. Oberwolfach Reports 8.1, 609–638.
Awodey, S. and Warren, M. A. (2009) Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (45) 4555. ArXiv:0709.0248.
Barthel, T. and Riehl, E. (2013) On the construction of functorial factorizations for model categories. Algebraic and Geometric Topology 13 10891124. ArXiv:1204.5427.
Bellissima, F. (1986) Finitely generated free Heyting algebras. Journal of Symbolic Logic 51 (1) 152165.
Berger, C. and Moerdijk, I. (2011) On an extension of the notion of Reedy category. Mathematische Zeitschrift 269 (3) 9771004. ArXiv:0809.3341.
Bergner, J. E. and Rezk, C. (2013) Reedy categories and the Θ-construction. Mathematische Zeitschrift 274 (1–2) 499514. ArXiv:1110.1066.
Brown, K. S. (1974) Abstract homotopy theory and generalized sheaf cohomology. Transactions of the American Mathematical Society 186 419458.
Cartmell, J. (1986) Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 209243.
Cisinski, D.-C. (2002) Théories homotopiques dans les topos. Journal of Pure and Applied Algebra 174 4382.
Cisinski, D.-C. (2006) Les préfaisceaux comme modèles type d'homotopie. Vol. 308. Astérisque. Soc. Math. France.
Cisinski, D.-C. (2012) Blog comment on post The mysterious nature of right properness. Available at: http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306.
Gambino, N. and Garner, R. (2008) The identity type weak factorisation system. Theoretical Computer Science 409 (1) 94109.
Gepner, D. and Kock, J. (2012) Univalence in locally Cartesian closed 1-categories. ArXiv:1208.1749.
Hedberg, M. (1998) A coherence theorem for Martin–Löf's type theory. Journal of Functional Programming 8 (4) 413436.
Hirschhorn, P. S. (2003) Model Categories and their Localizations, Mathematical Surveys and Monographs volume 99, American Mathematical Society.
Hofmann, M. (1994) On the interpretation of type theory in locally cartesian closed categories. In: Proceedings of Computer Science Logic. Springer Lecture Notes in Computer Science 427–441.
Hofmann, M. and Streicher, T. (1998) The groupoid interpretation of type theory. In: Twenty-five years of constructive type theory (Venice, 1995). Oxford Logic Guides, volume 36, New York: Oxford University Press 83111.
Hofstra, P. and Warren, M. A. (2013) Combinatorial realizability models of type theory. Annals of Pure and Applied Logic 164 (10) 957988. ArXiv:1205.5527.
HoTT Project (2013) The homotopy type theory coq library. Available at: http://github.com/HoTT/HoTT/.
Hovey, M. (1999) Model Categories, Mathematical Surveys and Monographs volume 63, American Mathematical Society.
Jacobs, B. (1999) Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics volume 141, Amsterdam: North Holland.
Johnson, M. W. (2010) On modified Reedy and modified projective model structures. Theory and Applications of Categories 24 (8) 179208.
Kapulkin, C., Lumsdaine, P. L. and Voevodsky, V. (2012) The simplicial model of univalent foundations. ArXiv:1211.2851.
Licata, D. R. and Harper, R. (2012) Canonicity for 2-dimensional type theory. In: Proceedings of the 39th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. POPL '12, Philadelphia, PA, USA: ACM 337348.
Lumsdaine, P. L. (2010) Weak omega-categories from intensional type theory. Typed Lambda Calculi and Applications 6 119. ArXiv:0812.0409.
Lumsdaine, P. L. (2011) Strong functional extensionality from weak. Available at: http://homotopytypetheory.org/2011/12/19/strong-funext-from-weak/.
Lumsdaine, P. L. and Shulman, M. (2014) Semantics of higher inductive types. In preparation.
Lumsdaine, P. L. and Warren, M. (2014) The local universes model: an overlooked coherence construction for dependent type theories. ArXiv:1411.1736.
Lurie, J. (2009) Higher Topos Theory, Annals of Mathematics Studies volume 170, Princeton University Press. ArXiv:math.CT/0608040.
Makkai, M. (1995) First order logic with dependent sorts, with applications to category theory. Available at: http://www.math.mcgill.ca/makkai/folds/.
Moerdijk, I. (2012) Fiber bundles and univalence. Available at: http://www.pitt.edu/~krk56/fiber_bundles_univalence.pdf. (Notes prepared by Chris Kapulkin).
Quillen, D. G. (1967) Homotopical Algebra, Lecture Notes in Mathematics volume 43, Springer-Verlag.
Radulescu-Banu, A. (2006) Cofibrations in homotopy theory. ArXiv:math/0610009.
Reedy, C. L. (n.d.) Homotopy theory of model categories. Available at: http://www-math.mit.edu/~psh/.
Shulman, M. (2014) The univalence axiom for elegant Reedy presheaves. To appear in HHA. ArXiv:1307.6248.
Streicher, T. (1991) Semantics of Type Theory: Correctness, Completeness, and Independence Results, Progress in Theoretical Computer Science, Birkhaäuser.
Strøm, A. (1972) The homotopy category is a homotopy category. Archiv der Mathematik (Basel) 23 435441.
Univalent Foundations Program (2013) Homotopy type theory: Univalent foundations of mathematics. Available at: http://homotopytypetheory.org/book/.
van den Berg, B. and Garner, R. (2011) Types are weak ω-groupoids. Proceedings of the London Mathematical Society 102 (2) 370394.
van den Berg, B. and Garner, R. (2012) Topological and simplicial models of identity types. ACM Transactions on Computational Logic 13 (1) 3:13:44.
Voevodsky, V. (2011) Notes on type systems. Available at: http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html.
Voevodsky, V. (2013) Univalent foundations. Avaiolable at: http://github.com/vladimirias/Foundations/.
Wadler, P. (1989) Theorems for free! In: Functional Programming Languages and Computer Architecture, ACM Press 347359.
Warren, M. A. (2008) Homotopy Theoretic Aspects of Constructive Type Theory, Ph.D. thesis, Carnegie Mellon University.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed