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An étalé space construction for stacks

David Carchedi

Algebraic & Geometric Topology 13 (2013) 831–903

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.

étalé space, étale stack, groupoid, topological stack, differentiable stack, action groupoid, topos, topoi
Mathematical Subject Classification 2010
Primary: 22A22, 58H05, 53C08
Secondary: 18B25, 14A20, 18F20
Received: 24 April 2012
Revised: 16 October 2012
Accepted: 28 October 2012
Published: 30 March 2013
David Carchedi
Max Planck Institute for Mathematics
Vivatsgasse 7
D-53113 Bonn