A Homotopy Theory for Stacks

Abstract

We give a homotopy theoretic characterization of stacks on a site $\cC$ as the {\it homotopy sheaves} of groupoids on $\cC$. 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 $\cC$.

24 Pages. Much abridged for publication. To appear in Israel Journal of Math

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• The Grothendieck Construction and Gradings for Enriched Categories
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• Cylindrical model structures
• Abelian Groups in omega-categories