An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid. Morita equivalent Lie groupoids give rise to the same orbifold. One can consider a bicategory of proper étale Lie groupoids and the orbifolds will be the objects of certain bicategorical localization of this bicategory (a result of Moerdijk and Pronk).
An orbifold is traditionally defined as a topological space equipped with an orbifold structure, which is in turn an equivalence class of orbifold atlases. An orbifold is locally a stack quotient of a smooth manifold by a finite group. Every orbifold is globally a quotient of a smooth manifold by an action of finite-dimensional Lie group with finite stabilizers in each point (the construction uses the frame bundle). The word orbifold is invented by Thurston, while the original name was V-manifold (Satake), and was taken in a more restrictive sense, assuming that the actions of finite groups on the charts are always effective. Nowdays we call such orbifolds effective and those which are global quotients by a finite group global quotient orbifolds.
It has been noticed that the topological invariants of the underlying topological space of an orbifold as a topological space with an orbifold structure are not appropriate, but have to be corrected leading to orbifold Euler characteristics, orbifold cohomology etc. One of the constructions which is useful in this respect is the inertia orbifold (the inertia stack of the original orbifold) which gives rise to “twisted sectors” in Hilbert space of a quantum field theory on the orbifold, and also to twisted sectors in the appropriate cohomology spaces. A further generalization gives multitwisted sectors.
There is also a notion of finite stabilizers in algebraic geometry. A singular variety is called an (algebraic) orbifold if it has only so-called orbifold singularities.
Ieke Moerdijk, Orbifolds as Groupoids: an Introduction (arXiv blog)
Eugene Lerman, Orbifolds as stacks? (arXiv)
Orbifolds often appear as moduli spaces in differential geometric setting:
An interesting generalization of an orbifold is so-called weighted branched manifold; see
Wikipedia article is mainly tailored toward Thurston’s approach.
I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely does not view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.
Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?
Urs Schreiber: please, go ahead. It would be appreciated.