An orbifold is a paracompact Hausdorff space which can locally be described as the quotient of an open subset of $R^n$ by the action of a finite group [4]. For example, the orbit space of the action of a compact Lie group on a manifold, which acts with finite isotropy groups, is an orbifold. Orbifolds which can be obtained in this way are called representable.

In order to study orbifold homotopy theory, we will use a representation by Lie groupoids. Such representations are unique up to essential equivalence. An orbifold is representable precisely when it has a representation by a translation groupoid. We will describe a notion of generalized morphism between Lie groupoids, which takes this notion of essential equivalence into account. These morphisms correspond precisely to what Adem, Chen, Ruan, and others have called good maps of orbifolds. They have been used to study orbifold $K$-theory and various kinds of orbifold cohomology with applications to mathematical physics.

For representable orbifolds, one may also want to use the techniques of equivariant homotopy theory to obtain invariants, since such an orbifold is a $G$-space. However, this representation is not unique, one may be able to describe the same orbifold as both $M/G$ and as $N/H$. In this case the two translation groupoids would be essentially equivalent. So our goal will be to describe an orbifold Bredon cohomology which is invariant under essential equivalence.

In this talk I will give a description of generalized maps between translation groupoids in terms of equivariant morphisms, and give a precise characterization of the essential equivalences between such groupoids. As an application I will show that equivariant Bredon cohomology can be viewed as an orbifold invariant. In the special case when one takes Bredon cohomology with coefficients in the representation rings of the isotropy subgroups, one obtains orbifold $K$-theory. This was considered by Adem and Ruan [1] with coefficients tensored over the rational numbers, and by Honkasalo [3] over a general ground ring. However, our proof is more widely applicable, and much simpler than the proof by Adem.

1. A. Adem, Y. Ruan, Twisted orbifold $K$-theory, Comm. Math. Phys., 237 (2003), pp. 533-556.
2. W. Chen, Y. Ruan, A new cohomology theory of orbifold, Comm. Math. Phys., 248 (2004), pp. 1-31.
3. H. Honkasalo, Equivariant Alexander-Spanier cohomology for actions of compact Lie groups, Math. Scand., 67 (1990), pp. 23-34.
4. I. Satake, On a generalization of the notion of manifold, Proc. of the Nat. Acad. of Sc. U.S.A., 42 (1956), pp. 359-363.