Geometry & Topology

Comparison of models for $(\infty, n)$–categories, I

Julia E Bergner and Charles Rezk

Full-text: Open access

Abstract

While many different models for (,1)–categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for (,n)–categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as (,n)–categories. Furthermore, we establish Quillen equivalences between them.

Article information

Source
Geom. Topol., Volume 17, Number 4 (2013), 2163-2202.

Dates
Received: 17 April 2012
Revised: 18 March 2013
Accepted: 17 April 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732649

Digital Object Identifier
doi:10.2140/gt.2013.17.2163

Mathematical Reviews number (MathSciNet)
MR3109865

Zentralblatt MATH identifier
1273.18031

Subjects
Primary: 55U40: Topological categories, foundations of homotopy theory
Secondary: 55U35: Abstract and axiomatic homotopy theory 18D15: Closed categories (closed monoidal and Cartesian closed categories, etc.) 18D20: Enriched categories (over closed or monoidal categories) 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 18C10: Theories (e.g. algebraic theories), structure, and semantics [See also 03G30]

Keywords
$(\infty, n)$–categories $\Theta_n$–spaces enriched categories

Citation

Bergner, Julia E; Rezk, Charles. Comparison of models for $(\infty, n)$–categories, I. Geom. Topol. 17 (2013), no. 4, 2163--2202. doi:10.2140/gt.2013.17.2163. https://projecteuclid.org/euclid.gt/1513732649


Export citation

References

  • D Ayala, N Rozenblyum, work in progress
  • B Badzioch, Algebraic theories in homotopy theory, Ann. of Math. 155 (2002) 895–913
  • C Barwick, Homotopy coherent algebra II: Iterated wreath products of $O$ and $(\infty, n)$–categories, In preparation
  • C Barwick, C Schommer-Pries, On the unicity of the homotopy theory of higher categories
  • T Beke, Sheafifiable homotopy model categories, II, J. Pure Appl. Algebra 164 (2001) 307–324
  • C Berger, Iterated wreath product of the simplex category and iterated loop spaces, Adv. Math. 213 (2007) 230–270
  • J E Bergner, Rigidification of algebras over multi-sorted theories, Algebr. Geom. Topol. 6 (2006) 1925–1955
  • J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043–2058
  • J E Bergner, Simplicial monoids and Segal categories, 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) 59–83
  • J E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397–436
  • J E Bergner, A survey of $(\infty,1)$–categories, from: “Towards higher categories”, IMA Vol. Math. Appl. 152, Springer, New York (2010) 69–83
  • J E Bergner, C Rezk, Enriched Segal categories, In preparation
  • J E Bergner, C Rezk, Reedy categories and the $\varTheta$–construction, Math. Z. 274 (2013) 499–514
  • D Dugger, D I Spivak, Mapping spaces in quasicategories, Algebr. Geom. Topol. 11 (2011) 263–325
  • W G Dwyer, J Spaliński, Homotopy theories and model categories, from: “Handbook of algebraic topology”, (I M James, editor), North-Holland, Amsterdam (1995) 73–126
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
  • P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)
  • A Hirschowitz, C Simpson, Descente pour les $n$–champs
  • M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999)
  • A Joyal, Simplicial categories vs quasi-categories, In preparation
  • A Joyal, The theory of quasi-categories I, In preparation
  • 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 Jurie, $(\infty, 2)$–categories and Goodwillie calculus
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
  • J Lurie, On the classification of topological field theories, from: “Current developments in mathematics, 2008”, (D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S-T Yau, editors), Int. Press, Somerville, MA (2009) 129–280
  • S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer, New York (1998)
  • R Pellissier, Catégories enrichies faibles
  • D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer, Berlin (1967)
  • C L Reedy, Homotopy theory of model categories, Unpublished manuscript Available at \setbox0\makeatletter\@url http://www-math.mit.edu/~psh/reedy.pdf {\unhbox0
  • C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973–1007
  • C Rezk, A Cartesian presentation of weak $n$–categories, Geom. Topol. 14 (2010) 521–571
  • C Simpson, Homotopy theory of higher categories, New Mathematical Monographs 19, Cambridge Univ. Press (2012)
  • B Toën, Vers une axiomatisation de la théorie des catégories supérieures, $K$–Theory 34 (2005) 233–263
  • D R B Verity, Weak complicial sets, I: Basic homotopy theory, Adv. Math. 219 (2008) 1081–1149