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 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.

v2: minor changes

References in corpus (3)

• Orbifolds as Groupoids: an Introduction
• Topological and Smooth Stacks
• On the space of morphisms between étale groupoids

Cited by in corpus (6)

• An Invitation to Higher Gauge Theory
• Internal categories, anafunctors and localisations
• Four Equivalent Versions of Non-Abelian Gerbes
• Moduli of Open Stable Maps to a Homogeneous Space
• The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces
• A bicategory of reduced orbifolds from the point of view of differential geometry - I