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Characterizing Artin stacks
Authors:Sharon Hollander
Affiliation:1. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001, Lisbon, Portugal
Abstract:We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site ></img> </span> . There is a natural method for extending a property <em>P</em> of morphisms of sheaves on <span class= ></img> </span> to a property <span class= ${mathcal{P}}$ of morphisms of presheaves of groupoids. We prove that the property ${mathcal{P}}$ is homotopy invariant in the local model structure on ></img> </span> when <em>P</em> is stable under pullback and local on the target. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, we obtain homotopy theoretic characterizations of algebraic and Artin stacks as those which are equivalent to simplicial objects in <span class= ></img> </span> satisfying certain analogues of the Kan conditions. The definition of Artin stack can naturally be placed within a hierarchy which roughly measures how far a stack is from being representable. We call the higher analogues of Artin stacks <em>n</em>-<em>algebraic stacks</em>, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, <em>n</em>-algebraic is the same as 3-algebraic for all <em>n</em> ≥ 3. As an application of these results we show that a stack is <em>n</em>-algebraic if and only if the homotopy orbits of a group action on it is.</td> </tr> <tr> <td align=
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