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Forum Mathematicum

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Volume 29, Issue 3


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
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Published Online: 2016-11-06 | DOI: https://doi.org/10.1515/forum-2015-0228


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


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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.

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