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A homotopy theory for stacks

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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 S 2-nullification of Jardine’s model structure on sheaves of simplicial sets on C.

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References

  1. D. W. Anderson, Fibrations and Geometric Realizations, Bulletin of American Mathematical Society 84 (1978), 765–786.

    Article  MATH  Google Scholar 

  2. A. K. Bousfield, Homotopy Spectral Sequences and Obstructions, Israel Journal of Mathematics 66 (1989), 54–105.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics 304, Springer-Verlag, Berlin-New York, 1972

    MATH  Google Scholar 

  4. L. Breen, On the Classification of 2-Gerbes and 2-Stacks, Asterisque 225 (1994).

  5. 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.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. J. Dubuc, Kan Extensions in Enriched Category Theory, Lecture Notes in Mathematics 145, Springer-Verlag, Berlin-New York, 1970.

    MATH  Google Scholar 

  7. D. Dugger, Universal homotopy theories, Advances in Mathematics 164 (2001), 144–176.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Dugger, S. Hollander and D. Isaksen, Hypercovers and Simplicial Presheaves, Mathematical Proceedings of the Cambridge Philosophical Society 136 (2004), 9–51.

    Article  MATH  MathSciNet  Google Scholar 

  9. W. G. Dwyer and D. M. Kan, Homotopy Commutative Diagrams and their Realizations, Journal of Pure and Applied Algebra 57 (1989), 5–24.

    Article  MATH  MathSciNet  Google Scholar 

  10. 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. 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. E. Dror Farjoun, CCellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics 1635, Springer-Verlag, Berlin-New York, 1995.

    Google Scholar 

  13. P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer-Verlag, New York, 1967.

    MATH  Google Scholar 

  14. J. Giraud, Cohomologie Non-Abelienne, Springer-Verlag, Berlin-Heidelberg-New York, 1971.

    MATH  Google Scholar 

  15. P. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics 174, Birkhäuser, Basel, 1999.

    MATH  Google Scholar 

  16. P. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs, 99, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  17. A. Hirschowitz and C. Simpson, Descente pour les n-champs, see the archive listing: math.AG/9807049

  18. S. Hollander, A Homotopy Theory for Stacks, PhD Thesis, MIT, 2001.

  19. M. Hovey, Model Categories, Mathematical Surveys and Monographs 63, American Mathematical Society, 1999.

  20. J. F. Jardine, Simplicial Presheaves, Journal of Pure and Applied Algebra 47 (1987), 35–87.

    Article  MATH  MathSciNet  Google Scholar 

  21. 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. S. MacLane, Categories for the Working Mathematician, aduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1971.

    Google Scholar 

  23. 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. J. P. May, Simplicial objects in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill., 1969.

    Google Scholar 

  25. D. J. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springer-Verlag, Berlin-New York, 1967.

    MATH  Google Scholar 

  26. J. Smith, Combinatorial Model Categories, preprint.

  27. N. Strickland, K(n)-local duality for finite groups and groupoids, Topology 39 (2000), 1021–1033.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. Tamme, Introduction to Etale Cohomology, Springer-Verlag, Berlin, 1994.

    MATH  Google Scholar 

  29. B. Toen and G. Vezzosi, Homotopical Algebraic Geometry I: Topos Theory, Advances in Mathematics 193 (2005), 257–372.

    Article  MATH  MathSciNet  Google Scholar 

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Author notes

  1. Sharon Hollander

    Present address: Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Mathema’tica, Instituto Superior Te’cnico, 1049-001, Lisboa, Portugal

Authors and Affiliations

  1. Department of Mathematics, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem Jerusalem, 91904, Israel

    Sharon Hollander

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  1. Sharon Hollander
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Correspondence to Sharon Hollander.

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Hollander, S. A homotopy theory for stacks. Isr. J. Math. 163, 93–124 (2008). https://doi.org/10.1007/s11856-008-0006-5

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  • DOI: https://doi.org/10.1007/s11856-008-0006-5

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