Orbifolds as stacks?

Details Orbifolds as stacks? Orbifolds as stacks?

2008-06-25

arxiv.org/abs/0806.4160v2

L'Enseign. Math. (2) 56 (2010), no. 3-4, 315--363

Mathematics

Differential Geometry

v2: minor changes

Scientific paper

The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" definition of orbifolds as proper etale Lie groupoids up to Morita equivalence. The second goal is to describe two complementary ways of thinking of orbifolds as a 2-category: 1. the weak 2-category of foliation Lie groupoids, bibundles and equivariant maps between bibundles and 2. the strict 2-category of Deligne-Mumford stacks over the category of smooth manifolds.

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