Homotopy Theory of Orbispaces

TL;DR

This paper extends the homotopy theory equivalence from G-spaces to orbispaces by developing a new categorical framework involving topological groups and groupoids, with model category structures.

Contribution

It generalizes Elmendorf's theorem to orbispaces, introducing a new categorical setting with model structures for topological groupoids.

Findings

Established a model category structure on topological groupoids.

Extended the equivalence of homotopy theories to orbispaces.

Provided new tools for studying orbispaces via topological groupoids.

Abstract

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.