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Univalence in locally cartesian closed ∞-categories

  • David Gepner and Joachim Kock EMAIL logo
Published/Copyright: November 6, 2016
Forum Mathematicum
From the journal Forum Mathematicum Volume 29 Issue 3

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 0≤n≤∞. 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

Communicated by Frederick R. Cohen

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

Funding statement: 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).

Acknowledgements

This work was originally prompted by the preprints of Kapulkin, Lumsdaine and Voevodsky [18], and Shulman [27]. We have also benefited from conversations with Peter Arndt, Steve Awodey, Nicola Gambino, André Joyal, Urs Schreiber, Markus Spitzweck, and especially Mike Shulman. We thank the anonymous referees for pertinent remarks that helped improve the exposition.

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Received: 2015-11-10
Revised: 2016-7-6
Published Online: 2016-11-6
Published in Print: 2017-5-1

© 2017 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
  3. The Selberg trace formula as a Dirichlet series
  4. On the cohomology and their torsion of real toric objects
  5. Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
  6. Non-abelian tensor and exterior products of multiplicative Lie rings
  7. On the infimum of the spectrum of a relativistic Schrödinger operator
  8. Character correspondences for symmetric groups and wreath products
  9. Univalence in locally cartesian closed ∞-categories
  10. On subordinate random walks
  11. Tame combings and easy groups
  12. On Belk’s classifying space for Thompson’s group F
  13. Subspace confinement for switched linear systems
  14. Exceptional bundles of homological dimension ${k}$
  15. Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
  16. Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
https://doi.org/10.1515/forum-2015-0228
Keywords for this article
Univalence; infinity-categories; infinity-topoi; infinity-quasitopoi; factorization systems,localization

Articles in the same Issue

  1. Frontmatter
  2. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
  3. The Selberg trace formula as a Dirichlet series
  4. On the cohomology and their torsion of real toric objects
  5. Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
  6. Non-abelian tensor and exterior products of multiplicative Lie rings
  7. On the infimum of the spectrum of a relativistic Schrödinger operator
  8. Character correspondences for symmetric groups and wreath products
  9. Univalence in locally cartesian closed ∞-categories
  10. On subordinate random walks
  11. Tame combings and easy groups
  12. On Belk’s classifying space for Thompson’s group F
  13. Subspace confinement for switched linear systems
  14. Exceptional bundles of homological dimension ${k}$
  15. Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
  16. Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
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