caa a22 4500 445836989 CHVBK 20180317145332.0 cr unu---uuuuu 170323e20111001xx s 000 0 eng 10.1007/s00209-010-0746-x doi (NATIONALLICENCE)springer-10.1007/s00209-010-0746-x Hollander Sharon Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001, Lisbon, Portugal aut Characterizing Artin stacks [Elektronische Daten] [Sharon Hollander] We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site . There is a natural method for extending a property P of morphisms of sheaves on to a property $${\mathcal{P}}$$ of morphisms of presheaves of groupoids. We prove that the property $${\mathcal{P}}$$ is homotopy invariant in the local model structure on when P 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 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 n-algebraic stacks, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, n-algebraic is the same as 3-algebraic for all n≥ 3. As an application of these results we show that a stack is n-algebraic if and only if the homotopy orbits of a group action on it is. Springer-Verlag, 2010 Mathematische Zeitschrift Springer-Verlag 269/1-2(2011-10-01), 467-494 0025-5874 269:1-2<467 2011 269 209 https://doi.org/10.1007/s00209-010-0746-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00209-010-0746-x text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Hollander Sharon Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001, Lisbon, Portugal aut NATIONALLICENCE 773 0- Mathematische Zeitschrift Springer-Verlag 269/1-2(2011-10-01), 467-494 0025-5874 269:1-2<467 2011 269 209 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer