A Survey of (∞, 1)-Categories

  • Julia E. Bergner
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 152)


In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible for n > 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.


Model Category Homotopy Theory Simplicial Category Weak Equivalence Left Adjoint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



I would like to thank Bill Dwyer, Chris Douglas, André Joyal, Jacob Lurie, Peter May, and Bertrand Toën for reading early drafts of this paper, making suggestions, and sharing their work in this area.


  1. [1]
    J.E. Bergner, A characterization of fibrant Segal categories, Proc. Amer. Math. Soc. (2007), 135:4031–4037.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J.E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. (2007), 359:2043–2058.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J.E. Bergner, Three models for the homotopy theory of homotopy theories, Topology (2007), 46:397–436.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J.M. Boardman and R.M. Vogt, Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, 347, Springer-Verlag, 1973.Google Scholar
  5. [5]
    J.M. Cordier and T. Porter, Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. (1986), 100:65–90.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    C.L. Douglas, Twisted parametrized stable homotopy theory, preprint available at math.AT/0508070.Google Scholar
  7. [7]
    D. Dugger, Universal homotopy theories, Adv. Math. (2001), 164(1):144–176.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    W.G. Dwyer and D.M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra (1980), 18:17–35.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    W.G. Dwyer and D.M. Kan, Equivalences between homotopy theories of diagrams, Algebraic topology and algebraic K-theory (Princeton, N.J.), 1983, 180–205, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton, NJ, 1987.Google Scholar
  10. [10]
    W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topology (1980), 19:427-440.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra (1980), 17(3):267–284.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    W.G. Dwyer, D.M. Kan, and J.H. Smith, Homotopy commutative diagrams and their realizations. J. Pure Appl. Algebra (1989), 57:5–24.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of Algebraic Topology, Elsevier, 1995.Google Scholar
  14. [14]
    P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math, Vol. 174, Birkhauser, 1999.Google Scholar
  15. [15]
    P.S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys, and Monographs, 99, AMS, 2003.Google Scholar
  16. [16]
    A. Hirschowitz and C. Simpson, Descente pour les n-champs, preprint available at math.AG/9807049.Google Scholar
  17. [17]
    Mark Hovey, Model Categories, Mathematical Surveys and Monographs, 63. American Mathematical Society 1999.Google Scholar
  18. [18]
    A. Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra (2002), 17(5):207–222.CrossRefMathSciNetGoogle Scholar
  19. [19]
    A. Joyal, Simplicial categories vs quasi-categories, in preparation.Google Scholar
  20. [20]
    André Joyal, The theory of quasi-categories I, in preparation.Google Scholar
  21. [21]
    André Joyal and Myles Tierney, Quasi-categories vs Segal spaces, Contemp. Math. (2007), 431:277–326.Google Scholar
  22. [22]
    Jacob Lurie, Higher topos theory, preprint available at math.CT/0608040.Google Scholar
  23. [23]
    Saunders Mac Lane, Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics 5, Springer-Verlag, 1997.Google Scholar
  24. [24]
    Timothy Porter, S-categories, S-groupoids, Segal categories and quasicategories, preprint available at math.AT/0401274.Google Scholar
  25. [25]
    Daniel Quillen, Homotopical Algebra, Lecture Notes in Math 43, Springer-Verlag, 1967.Google Scholar
  26. [26]
    C.L. Reedy, Homotopy theory of model categories, unpublished manuscript, available at˜psh.
  27. [27]
    Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc., 353(3):973–1007.Google Scholar
  28. [28]
    R. Schwänzl and R.M. Vogt, Homotopy homomorphisms and the hammock localization, Bol. Soc. Mat. Mexicana (2) (1992), 37(1–2):431–448.MATHMathSciNetGoogle Scholar
  29. [29]
    Graeme Segal, Categories and cohomology theories, Topology (1974), 13:293–312.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    Carlos Simpson, A closed model structure for n-categories, internal Hom, n-stacks, and generalized Seifert-Van Kampen, preprint, available at math.AG/9704006.Google Scholar
  31. [31]
    Carlos Simpson, A Giraud-type characterization of the simplicial categories associated to closed model categories as infty-pretopoi, preprint available at math.AT/9903167.Google Scholar
  32. [32]
    Goncalo Tabuada, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C.R. Math. Acad. Sci. Paris (2005), 340(1):15–19.MATHMathSciNetGoogle Scholar
  33. [33]
    Z. Tamsamani, Sur les notions de n-categorie et n-groupoíde non-stricte via des ensembles multi-simpliciaux, preprint available at alg-geom/9512006.Google Scholar
  34. [34]
    Bertrand Toën, Higher and derived stacks: a global overview, preprint available at math.AG/0604504.Google Scholar
  35. [35]
    Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. (2007), 167(3):615–667.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Bertrand Toën, Homotopical and Higher Categorical Structures in Algebraic Geometry (A View Towards Homotopical Algebraic Geometry), preprint available at math.AG/0312262.Google Scholar
  37. [37]
    Bertrand Toën, Vers une axiomatisation de la théorie des catégories supérieures, K-Theory (2005), 34(3):233–263.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    Bertrand Toën and Gabriele Vezzosi, Remark on K-theory and S-categories, Topology (2004), 43(4):765–791.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaRiversideUSA

Personalised recommendations