Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 29, Issue 3

Issues

Univalence in locally cartesian closed ∞-categories

David Gepner / Joachim Kock
  • Corresponding author
  • Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-11-06 | DOI: https://doi.org/10.1515/forum-2015-0228

Abstract

After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed -categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every -topos has a hierarchy of “universal” univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved (and detected) by right adjoints to locally cartesian localizations, and use this to exhibit certain canonical univalent families in -quasitopoi (certain -categories of “separated presheaves”, introduced here). We also exhibit some more exotic examples of univalent families, illustrating that a univalent family in an n-topos need not be (n-2)-truncated, as well as some univalent families in the Morel–Voevodsky -category of motivic spaces, an instance of a locally cartesian closed -category which is not an n-topos for any 0n. Lastly, we show that any presentable locally cartesian closed -category is modeled by a combinatorial type-theoretic model category, and conversely that the -category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed. Under this correspondence, univalent families in presentable locally cartesian closed -categories correspond to univalent fibrations in combinatorial type-theoretic model categories.

Keywords: Univalence; infinity-categories; infinity-topoi; infinity-quasitopoi; factorization systems,localization

MSC 2010: 55U35; 18C50

References

  • [1]

    Arndt P. and Kapulkin K., Homotopy-theoretic models of type theory, Typed Lambda Calculi and Applications, Lecture Notes Comput. Sci. 6690, Springer, Heidelberg (2011), 45–60. Google Scholar

  • [2]

    Awodey S., Garner R., Martin-Löf P. and Voevodsky V., The Homotopy interpretation of constructive type theory, Report No. 11/2011, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, 2011. Google Scholar

  • [3]

    Awodey S. and Warren M., Homotopy theoretic models of identity types, Math. Proc. Cambridge Philos. Soc. 146 (2009), 45–55. Google Scholar

  • [4]

    Barr M. and Wells C., Toposes, Triples and Theories, Grundlehren Math. Wiss. 278, Springer, New York, 1985. Corrected reprint in Repr. Theory Appl. Categ. 12 (2005), 1–287, http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html.

  • [5]

    Carboni A., Janelidze G., Kelly G. M. and Paré R., On localization and stabilization for factorization systems, Appl. Categ. Structures 5 (1997), 1–58. Google Scholar

  • [6]

    Cassidy C., Hébert M. and Kelly G. M., Reflective subcategories, localizations and factorization systems, J. Aust. Math. Soc. Ser. A 38 (1985), 287–329. Google Scholar

  • [7]

    Cisinski D.-C., Théories homotopiques dans les topos, J. Pure Appl. Algebra 174 (2002), 43–82. Google Scholar

  • [8]

    Cisinski D.-C., Univalent universes for elegant models of homotopy types, preprint 2014, https://arxiv.org/abs/1406.0058.

  • [9]

    D.-C. Cisinski and M. Shulman , Entry at the n-Category Café, http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306. Google Scholar

  • [10]

    Dugger D. and Spivak D., Mapping spaces in quasi-categories, Algebr. Geom. Topol. 11 (2011), 263–325. Google Scholar

  • [11]

    Dwyer W. G. and Kan D. M., Homotopy theory and simplicial groupoids, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), 379–385. Google Scholar

  • [12]

    Gambino N. and Garner R., The identity type weak factorisation system, Theoret. Comput. Sci. 409 (2008), 94–109. Google Scholar

  • [13]

    Garner R. and Lack S., Grothendieck quasitoposes, J. Algebra 355 (2012), 111–127. Google Scholar

  • [14]

    Gepner D. and Haugseng R., Enriched -categories via non-symmetric -operads, Adv. Math. 279 (2015), 575–716. Google Scholar

  • [15]

    Hofmann M. and Streicher T., The groupoid interpretation of type theory, Twenty-Five Years of Constructive Type Theory, Oxford Logic Guides 36, Oxford University Press, Oxford (1998), 83–111. Google Scholar

  • [16]

    Joyal A., The theory of quasi-categories, Advanced Course on Simplicial Methods in Higher Categories. Vol. II, Quaderns 45, CRM Barcelona, Bellaterra (2008), 147–497. Google Scholar

  • [17]

    Kapulkin K. and Lumsdaine P. L., The simplicial model of univalent foundations (after Voevodsky), preprint 2012, https://arxiv.org/abs/1211.2851.

  • [18]

    Kapulkin K., Lumsdaine P. L. and Voevodsky V., Univalence in simplicial sets, preprint 2012, http://arxiv.org/abs/1203.2553.

  • [19]

    Lumsdaine P. L., Weak ω-categories from intensional type theory, Log. Methods Comput. Sci. 6 (2010), no. 3:24, 1–19. Google Scholar

  • [20]

    Lumsdaine P. L. and Warren M., The local universes model: An overlooked coherence construction for dependent type theories, ACM Trans. Comput. Log. 16 (2015), Article ID 23. Google Scholar

  • [21]

    Lurie J., Higher Topos Theory, Ann. of Math. Stud. 170, Princeton University Press, Princeton, 2009, available from http://www.math.harvard.edu/~lurie/.

  • [22]

    Lurie J., Higher algebra, available from http://www.math.harvard.edu/~lurie/.

  • [23]

    Mac Lane S. and Moerdijk I., Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer, New York, 1995. Google Scholar

  • [24]

    Morel F. and Voevodsky V., 𝐀1-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143. Google Scholar

  • [25]

    Rezk C., A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), 973–1007. Google Scholar

  • [26]

    Shulman M., The univalence axiom for elegant Reedy presheaves, Homology Homotopy Appl. 17 (2015), 81–106. Google Scholar

  • [27]

    Shulman M., The univalence axiom for inverse diagrams and homotopy canonicity, Math. Structures Comput. Sci. 25 (2015), 1203–1277. Google Scholar

  • [28]

    Spitzweck M. and Østvær P. A., Motivic twisted K-theory, Algebr. Geom. Topol. 12 (2012), 565–599. Google Scholar

  • [29]

    Streicher T., A model of type theory in simplicial sets: A brief introduction to Voevodsky’s homotopy type theory, J. Appl. Log. 12 (2014), 45–49. Google Scholar

  • [30]

    van den Berg B. and Garner R., Types are weak ω-groupoids, Proc. Lond. Math. Soc. (3) 102 (2011), 370–394. Google Scholar

  • [31]

    Voevodsky V., Notes on type systems, 2011, available from http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html.

  • [32]

    nLab entry, Model of type theory in an (infinity,1)-topos, https://ncatlab.org/homotopytypetheory/show/model+of+type+theory+in+an+(infinity,1)-topos.

  • [33]

    The Univalent Foundations Program, Homotopy Type Theory–Univalent Foundations of Mathematics, Institute for Advanced Study, Princeton, 2013, available from http://homotopytypetheory.org/book.

About the article


Received: 2015-11-10

Revised: 2016-07-06

Published Online: 2016-11-06

Published in Print: 2017-05-01


Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1406529

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: GE 2504/1-1

Funding Source: Ministerio de Ciencia e Innovación

Award identifier / Grant number: MTM2009-10359

Award identifier / Grant number: MTM2010-20692

Award identifier / Grant number: MTM2013-42293-P

The first author was partially supported by NSF grant DMS-1406529 and DFG grant GE 2504/1-1. The second author was partially supported by grants MTM2009-10359, MTM2010-20692 and MTM2013-42293-P of Spain and by SGR1092-2009 of AGAUR (Catalonia).


Citation Information: Forum Mathematicum, Volume 29, Issue 3, Pages 617–652, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0228.

Export Citation

© 2017 by De Gruyter.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
PETER LEFANU LUMSDAINE and MICHAEL SHULMAN
Mathematical Proceedings of the Cambridge Philosophical Society, 2019, Page 1
[2]
Matthew Ando, Andrew Blumberg, and David Gepner
Geometry & Topology, 2018, Volume 22, Number 7, Page 3761
[3]
Benno van den Berg and Ieke Moerdijk
Mathematische Annalen, 2017

Comments (0)

Please log in or register to comment.
Log in