Algebraic & Geometric Topology

Mapping spaces in quasi-categories

Daniel Dugger and David I Spivak

Full-text: Open access

Abstract

We apply the Dwyer–Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [Alg. Geom. Topol. 11 (2011) 225–261], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 1 (2011), 263-325.

Dates
Received: 22 December 2009
Revised: 16 August 2010
Accepted: 27 September 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715187

Digital Object Identifier
doi:10.2140/agt.2011.11.263

Mathematical Reviews number (MathSciNet)
MR2764043

Zentralblatt MATH identifier
1214.55013

Subjects
Primary: 55U40: Topological categories, foundations of homotopy theory
Secondary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 18B99: None of the above, but in this section

Keywords
quasi-category infinity category Dwyer–Kan mapping space simplicial category Joyal model structure homotopy function complex

Citation

Dugger, Daniel; Spivak, David I. Mapping spaces in quasi-categories. Algebr. Geom. Topol. 11 (2011), no. 1, 263--325. doi:10.2140/agt.2011.11.263. https://projecteuclid.org/euclid.agt/1513715187


Export citation

References

  • T Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000) 447–475
  • J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043–2058
  • J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer, Berlin (1973)
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, Berlin (1972)
  • J-M Cordier, T Porter, Homotopy coherent category theory, Trans. Amer. Math. Soc. 349 (1997) 1–54
  • D Dugger, Classification spaces of maps in model categories
  • D Dugger, A primer on homotopy colimits, Preprint (2008) Available at \setbox0\makeatletter\@url http://math.uoregon.edu/~ddugger/hocolim.pdf {\unhbox0
  • D Dugger, D I Spivak, Rigidification of quasi-categories, Alg. Geom. Topol. 11 (2011) 225–261
  • W G Dwyer, D M Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980) 17–35
  • W G Dwyer, D M Kan, Function complexes in homotopical algebra, Topology 19 (1980) 427–440
  • W G Dwyer, D M Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267–284
  • P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003)
  • A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207–222 Special volume celebrating the $70$–th birthday of Professor Max Kelly
  • A Joyal, Simplicial categories vs. quasi-categories, Preprint (2008)
  • A Joyal, The theory of quasi-categories, Preprint (2008)
  • J Lurie, Higher topos theory, Annals of Math. Studies 170, Princeton Univ. Press (2009)
  • J P Nichols-Barrer, On quasi-categories as a foundation for higher algebraic stacks, PhD thesis, Massachusetts Institute of Technology (2007) Available at \setbox0\makeatletter\@url http://proquest.umi.com/pqdlink?FMT=7&DID=1372017801&RQT=309 {\unhbox0