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.
Unable to display preview. Download preview PDF.
- [Brn]L. Breen, On the Classification of 2-Gerbes and 2-Stacks, Asterisque 225 (1994).Google Scholar
- [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
- [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
- [DF]E. Dror Farjoun, CCellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics 1635, Springer-Verlag, Berlin-New York, 1995.Google Scholar
- [HS]A. Hirschowitz and C. Simpson, Descente pour les n-champs, see the archive listing: math.AG/9807049Google Scholar
- [Holl]S. Hollander, A Homotopy Theory for Stacks, PhD Thesis, MIT, 2001.Google Scholar
- [Ho]M. Hovey, Model Categories, Mathematical Surveys and Monographs 63, American Mathematical Society, 1999.Google Scholar
- [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
- [Ml]S. MacLane, Categories for the Working Mathematician, aduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1971.Google Scholar
- [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
- [May]J. P. May, Simplicial objects in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill., 1969.Google Scholar
- [Sm]J. Smith, Combinatorial Model Categories, preprint.Google Scholar