Orbifolds as Groupoids: an Introduction

The purpose of this paper is to describe orbifolds in terms of (a certain kind of) groupoids. In doing so, I hope to convince you that the theory of (Lie) groupoids provides a most convenient language for developing the foundations of the theory of orbifolds. Indeed, rather than defining all kinds of structures and invariants in a seemingly ad hoc way, often in terms of local charts, one can use groupoids (which are global objects) as a bridge from orbifolds to standard structures and invariants. This applies e.g. to the homotopy type of an orbifold (via the classifying space of the groupoid), the K-theory of an orbifold (via the equivariant vector bundles over the groupoid), the sheaf cohomology of an orbifold (via the derived category of equivariant sheaves over the groupoid), and many other such notions. Groupoids also help to clarify the relation between orbifolds and complexes of groups. Furthermore, the relation between orbifolds and non-commutative geometry is most naturally explained in terms of the convolution algebra of the groupoid. In this context, and in several others, the inertia groupoid of a given groupoid makes its natural appearance, and helps to explain how Bredon cohomology enters the theory of orbifolds.