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 étale 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.
Cite this article
Eugene Lerman, Orbifolds as stacks?. Enseign. Math. 56 (2010), no. 3, pp. 315–363