  Given a topological group G, its orbit category Orb_G has the transitive
G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A
well known theorem of Elmendorf then states that the category of G-spaces and
the category of contravariant functors Func(Orb_G,Spaces) have equivalent
homotopy theories. We extend this result to the context of orbispaces, with the
role of Orb_G now played by a category whose objects are topological groups and
whose morphisms are given by Hom(H,G) = Mono(H,G) x_G EG. On our way, we endow
the category of topological groupoids with notions of weak equivalence, fibrant
objects, and cofibrant objects, and show that it then shares many of the
properties of a Quillen model category.
