Israel Journal of Mathematics

, Volume 163, Issue 1, pp 93–124 | Cite as

A homotopy theory for stacks

Article

Abstract

We give a homotopy theoretic characterization of stacks on a site C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the S2-nullification of Jardine’s model structure on sheaves of simplicial sets on C.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [An]
    D. W. Anderson, Fibrations and Geometric Realizations, Bulletin of American Mathematical Society 84 (1978), 765–786.MATHCrossRefGoogle Scholar
  2. [Bo]
    A. K. Bousfield, Homotopy Spectral Sequences and Obstructions, Israel Journal of Mathematics 66 (1989), 54–105.MATHCrossRefMathSciNetGoogle Scholar
  3. [BK]
    A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics 304, Springer-Verlag, Berlin-New York, 1972MATHGoogle Scholar
  4. [Brn]
    L. Breen, On the Classification of 2-Gerbes and 2-Stacks, Asterisque 225 (1994).Google Scholar
  5. [DM]
    P. Deligne and D. Mumford, The Irreducibility of the Space of Curves of Given Genus, Publications Mathématiques de l’Institut des Hautes Études Scientifiques 36 (1969), 75–110.MATHCrossRefMathSciNetGoogle Scholar
  6. [Db]
    E. J. Dubuc, Kan Extensions in Enriched Category Theory, Lecture Notes in Mathematics 145, Springer-Verlag, Berlin-New York, 1970.MATHGoogle Scholar
  7. [Dg]
    D. Dugger, Universal homotopy theories, Advances in Mathematics 164 (2001), 144–176.MATHCrossRefMathSciNetGoogle Scholar
  8. [DHI]
    D. Dugger, S. Hollander and D. Isaksen, Hypercovers and Simplicial Presheaves, Mathematical Proceedings of the Cambridge Philosophical Society 136 (2004), 9–51.MATHCrossRefMathSciNetGoogle Scholar
  9. [DK]
    W. G. Dwyer and D. M. Kan, Homotopy Commutative Diagrams and their Realizations, Journal of Pure and Applied Algebra 57 (1989), 5–24.MATHCrossRefMathSciNetGoogle Scholar
  10. [DS]
    W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of Algebraic Topology, Elsevier, Amsterdam, 1995, pp. 73–126.Google Scholar
  11. [EK]
    S. Eilenberg and G. M. Kelly, Closed categories, in Proc. Conf. on Categorical Algebra (La Jolla 1965), Springer, New York, 1966, pp. 421–562.Google Scholar
  12. [DF]
    E. Dror Farjoun, CCellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics 1635, Springer-Verlag, Berlin-New York, 1995.Google Scholar
  13. [GZ]
    P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer-Verlag, New York, 1967.MATHGoogle Scholar
  14. [Gi]
    J. Giraud, Cohomologie Non-Abelienne, Springer-Verlag, Berlin-Heidelberg-New York, 1971.MATHGoogle Scholar
  15. [GJ]
    P. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics 174, Birkhäuser, Basel, 1999.MATHGoogle Scholar
  16. [Hi]
    P. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs, 99, American Mathematical Society, Providence, RI, 2003.MATHGoogle Scholar
  17. [HS]
    A. Hirschowitz and C. Simpson, Descente pour les n-champs, see the archive listing: math.AG/9807049Google Scholar
  18. [Holl]
    S. Hollander, A Homotopy Theory for Stacks, PhD Thesis, MIT, 2001.Google Scholar
  19. [Ho]
    M. Hovey, Model Categories, Mathematical Surveys and Monographs 63, American Mathematical Society, 1999.Google Scholar
  20. [Ja]
    J. F. Jardine, Simplicial Presheaves, Journal of Pure and Applied Algebra 47 (1987), 35–87.MATHCrossRefMathSciNetGoogle Scholar
  21. [JT]
    A. Joyal and M. Tierney, Strong stacks and classifying spaces. in Category theory (Como, 1990), Lecture Notes in Mathematics, 1488, Springer, Berlin, 1991, pp. 213–236.Google Scholar
  22. [Ml]
    S. MacLane, Categories for the Working Mathematician, aduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1971.Google Scholar
  23. [MM]
    S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, Berlin Heidelberg New York, 1992.Google Scholar
  24. [May]
    J. P. May, Simplicial objects in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill., 1969.Google Scholar
  25. [Q]
    D. J. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springer-Verlag, Berlin-New York, 1967.MATHGoogle Scholar
  26. [Sm]
    J. Smith, Combinatorial Model Categories, preprint.Google Scholar
  27. [St]
    N. Strickland, K(n)-local duality for finite groups and groupoids, Topology 39 (2000), 1021–1033.MATHCrossRefMathSciNetGoogle Scholar
  28. [Ta]
    G. Tamme, Introduction to Etale Cohomology, Springer-Verlag, Berlin, 1994.MATHGoogle Scholar
  29. [TV]
    B. Toen and G. Vezzosi, Homotopical Algebraic Geometry I: Topos Theory, Advances in Mathematics 193 (2005), 257–372.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2008

Authors and Affiliations

  1. 1.Department of Mathematics, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat RamThe Hebrew University of Jerusalem JerusalemIsrael
  2. 2.Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Mathema’ticaInstituto Superior Te’cnicoLisboaPortugal

Personalised recommendations