Formal Lie groupoids
An orbispace is a space, particularly a topological stack, that is locally modeled on the homotopy quotient/action groupoid of a locally compact topological space by a rigid group action.
Orbispaces are to topological spaces what orbifolds are to manifolds.
Write for the global orbit category. Then its (∞,1)-presheaf (∞,1)-category is the (∞,1)-category of orbispaces. (Henriques-Gepner 07, Rezk 14, remark 1.5.1)
By the main theorem of (Henriques-Gepner 07) the (∞,1)-presheaves on the global orbit category are equivalently “cellular” topological stacks/topological groupoids (“orbispaces”), we might write this as
Relation to global equivariant homotopy theory
The global equivariant homotopy theory is the (∞,1)-category (or else its homotopy category) of (∞,1)-presheaves on the global equivariant indexing category
Here has as objects compact Lie groups and the (∞,1)-categorical hom-spaces , where on the right we have the fundamental (∞,1)-groupoid of the topological groupoid of group homomorphisms and conjugations.
The global orbit category is the non-full subcategory of the global equivariant indexing category on the faithful maps .
The central theorem of (Rezk 14) is that is the base (∞,1)-topos over the cohesion of the slice of the global equivariant homotopy theory over the terminal orbispace (Rezk 14, p. 4 and section 7)
Rezk-global equivariant homotopy theory:
A detailed but elementary approach via atlases can be found in
and another approach is discussed in