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Abstract
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.
Keywords
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
Publication
Received: 24 April 2012
Revised: 16 October 2012
Accepted: 28 October 2012
Published: 30 March 2013