Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Artin M.: Versal deformations and algebraic stacks. Invent. Math. 27, 165–189 (1974)
Artin, M.: Algebraic spaces. James K. Whittemore Lecture in Mathematics given at Yale University, 1969. In: Yale Mathematical Monographs, vol. 3, vii+39 pp. Yale University Press, New Haven (1971)
Behrend, K.: PhD thesis. http://www.math.ubc.ca/~behrend/thesis.html
Borceux F.: Handbook of Categorical Algebra. University of Cambridge Press, Cambridge (1994)
Dugger, D., Isaksen, D.: Weak equivalences of simplicial presheaves. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory. Contemp. Math., vol. 346, pp. 97–113. American Mathematical Society
Dugger D., Hollander S., Isaksen D.: Hypercovers and simplicial presheaves. Math. Proc. Camb. Philos. Soc. 136(1), 9–51 (2004)
Deligne P., Mumford D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969)
Grothendieck, A.: Revêtements étales et groupe fondamental. In: Séminaire de Géométrie Algébrique du Bois Marie 1960–61 (SGA 1). Lecture Notes in Mathematics, vol. 224. Springer-Verlag, Berlin (1971)
Hollander S.: A Homotopy Theory for Stacks. Isr. J. Math. 163, 93–124 (2008)
Hollander S.: Characterizing algebraic stacks. Proc. Am. Math. Soc. 136(4), 1465–1476 (2008)
Hollander S.: Geometric criteria for Landweber exactness. Proc. Lond. Math. Soc. 99, 697–724 (2009)
Hollander S.: Diagrams indexed by Grothendieck constructions. Homol. Homotopy Appl. 10(3), 193–221 (2008)
Knutson, D.: Algebraic spaces. In: Lecture Notes in Mathematics, vol. 203, vi+261 pp. Springer-Verlag, Berlin (1971)
MacLane S., Moerdijk I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, Berlin Heidelberg New York (1992)
Pridham, J.P.: Presenting higher stacks as simplical schemes. arXiv:0905.4044
Ravenel, D.: Complex cobordism and stable homotopy groups of spheres. In: Pure and Applied Mathematics, vol. 121, xx+413 pp. Academic Press, Orlando (1986)
Simpson, C.: Algebraic (geometric) n-stacks. alg-geom/9609014
Toen B., Vezzosi G.: Homotopical algebraic geometry II: geometric stacks and applications. Mem. Am. Math. Soc. 193(902), x+224 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hollander, S. Characterizing Artin stacks. Math. Z. 269, 467–494 (2011). https://doi.org/10.1007/s00209-010-0746-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-010-0746-x