Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sheaf Theory for \'Etale Geometric Stacks   We generalize the notion of a small sheaf of sets over a topological space or
manifold to define the notion of a small stack of groupoids over an \'etale
topological or differentiable stack. We then provide a construction analogous
to the \'etal\'e space construction in this context, establishing an
equivalence of 2-categories between small stacks over an \'etale stack and
local homeomorphisms over it. We go on to characterize small sheaves and
gerbes. We show that ineffective data of \'etale stacks is completely described
by the theory of small gerbes. Furthermore, it is shown that \'etale stacks
(and in particular orbifolds) induce a small gerbe over their effective part,
and all gerbes arise in this way. It follows that ineffective orbifolds,
sometimes called non-reduced orbifolds, encode a canonical gerbe over their
effective (or reduced) part. For nice enough classes of maps, for instance
submersions, we show that \'etale stacks are equivalent to a 2-category of
gerbed effective \'etale stacks. Along the way, we also prove that the
2-category of topoi is a full reflective sub-2-category of localic stacks.

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The categories of this paper are: cs
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph