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.
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 orbit category (Henriques-Gepner 07, section 1.3), regarded as an (∞,1)-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.
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
(As such that global equivariant homotopy theory should be similar to ETop∞Grpd. Observe that this is a cohesive (∞,1)-topos with such that it sends a topological action groupoid of a topological group acting on a topological space to the homotopy quotient .)
The central theorem of (Rezk 14) (using a slightly different definition than Henriques-Gepner 07) is that is a cohesive (∞,1)-topos with producing homotopy quotients.
A detailed but elementary approach via atlases can be found in
and another approach is discussed in
Revised on March 7, 2014 02:05:51
by Urs Schreiber