An étalé space construction for stacks

Carchedi, David¹

¹Max Planck Institute for Mathematics, Vivatsgasse 7, D-53113 Bonn, Germany

Algebraic and Geometric Topology, Tome 13 (2013) no. 2, pp. 831-903

Résumé

We generalize the notion of a sheaf of sets over a space to define the notion of a small stack of groupoids over an étale stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of 2–categories between small stacks over an étale stack and local homeomorphisms over it. These results hold for a wide variety of types of spaces, for example, topological spaces, locales, various types of manifolds, and schemes over a fixed base (where stacks are taken with respect to the Zariski topology). Along the way, we also prove that the 2–category of topoi is fully reflective in the 2–category of localic stacks.

Détail

DOI : 10.2140/agt.2013.13.831

Classification : 22A22, 58H05, 53C08, 18B25, 14A20, 18F20
Keywords: étalé space, étale stack, groupoid, topological stack, differentiable stack, action groupoid, topos, topoi

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