Orbifolds as stacks?

Abstract

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.