Algebraic & Geometric Topology

Homotopy theory of cocomplete quasicategories

Karol Szumiło

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Abstract

We prove that the homotopy theory of cocomplete quasicategories is equivalent to the homotopy theory of cofibration categories. This is achieved by presenting both theories as fibration categories and constructing an explicit exact equivalence between them.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 2 (2017), 765-791.

Dates
Received: 29 June 2015
Revised: 30 April 2016
Accepted: 4 July 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1508431443

Digital Object Identifier
doi:10.2140/agt.2017.17.765

Mathematical Reviews number (MathSciNet)
MR3623671

Zentralblatt MATH identifier
1364.55019

Subjects
Primary: 55U35: Abstract and axiomatic homotopy theory
Secondary: 18G55: Homotopical algebra

Keywords
homotopy theory quasicategory homotopy colimit cofibration category

Citation

Szumiło, Karol. Homotopy theory of cocomplete quasicategories. Algebr. Geom. Topol. 17 (2017), no. 2, 765--791. doi:10.2140/agt.2017.17.765. https://projecteuclid.org/euclid.agt/1508431443


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References

  • C Barwick, D,M Kan, A characterization of simplicial localization functors and a discussion of DK equivalences, Indag. Math. 23 (2012) 69–79
  • C Barwick, D,M Kan, Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. 23 (2012) 42–68
  • C Barwick, C Schommer-Pries, On the unicity of the homotopy theory of higher categories, preprint (2013)
  • J,E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397–436
  • J,M Boardman, R,M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer, Berlin (1973)
  • K,S Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973) 419–458
  • D-C Cisinski, Catégories dérivables, Bull. Soc. Math. France 138 (2010) 317–393
  • D Dugger, D,I Spivak, Mapping spaces in quasi-categories, Algebr. Geom. Topol. 11 (2011) 263–325
  • W,G Dwyer, D,M Kan, J,H Smith, Homotopy commutative diagrams and their realizations, J. Pure Appl. Algebra 57 (1989) 5–24
  • P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergeb. Math. Grenzgeb., Springer, New York (1967)
  • P,G Goerss, J,F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
  • A Hirschowitz, C Simpson, Descente pour les $n$–champs, preprint (2001)
  • M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc., Providence, RI (1999)
  • A Joyal, The theory of quasi-categories and its applications, from “Advanced course on simplicial methods in higher categories”, Quaderns 45, CRM, Barcelona (2008) 149–496
  • A Joyal, M Tierney, Quasi-categories vs Segal spaces, from “Categories in algebra, geometry and mathematical physics” (A Davydov, M Batanin, M Johnson, S Lack, A Neeman, editors), Contemp. Math. 431, Amer. Math. Soc., Providence, RI (2007) 277–326
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton University Press (2009)
  • D,G Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer, Berlin (1967)
  • C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973–1007
  • E Riehl, D Verity, Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions, Homology Homotopy Appl. 17 (2015) 1–33
  • A Rădulescu-Banu, Cofibrations in homotopy theory, preprint (2009)
  • K Szumiło, Two models for the homotopy theory of cocomplete homotopy theories, PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2014) Available at \setbox0\makeatletter\@url http://hss.ulb.uni-bonn.de/2014/3692/3692.htm {\unhbox0
  • K Szumiło, Two models for the homotopy theory of cocomplete homotopy theories, preprint (2014)
  • K Szumiło, Frames in cofibration categories, J. Homotopy Relat. Struct. (2016) 1–40
  • K Szumiło, Homotopy theory of cofibration categories, Homology Homotopy Appl. 18 (2016) 345–357
  • B Toën, Vers une axiomatisation de la théorie des catégories supérieures, $K$–Theory 34 (2005) 233–263
  • R,M Vogt, The HELP-lemma and its converse in Quillen model categories, J. Homotopy Relat. Struct. 6 (2011) 115–118
  • F Waldhausen, Algebraic $K$–theory of spaces, from “Algebraic and geometric topology” (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 318–419