I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak Hausdorff or whatever) topological spaces with G-action and whose morphisms are G-maps, in which the generating cofibrations are maps of the form G/H x Sn-1 → G/H x Dn (n ≥ 0, H a closed subset of G) and the generating acyclic cofibrations are the obvious analogous thing. Apparently the weak equivalences in this category are those maps which induce weak equivalence on H-fixed points for every closed subgroup H of G. I assume the corresponding (∞,1)-category is presentable. (My preliminary question is, does anyone know a good source for this paragraph?)
My real question is: Can you give an (∞,1)-categorical description of this category, say via a universal property, or built somehow from the category of spaces? For instance, what is an explicit presentation as a localization of a category of presheaves of spaces? (An example of the kind of answer I am looking for is "functors from BG to Spaces", but that describes a model category of G-spaces whose weak equivalences are simply weak equivalences of the underlying spaces.)
(My next question would be asking for an analogous description of the equivariant stable homotopy category. I imagine this would be easy if I knew how to answer the first question, but if something special happens in the stable situation, I would like to know about it.)
Best Answer
I think a good reference for the first paragraph is "Equivariant Homotopy and Cohomology Theory" by Peter May and a bunch of other people. Chapter 5 includes "Elmendorf's theorem" that this homotopy theory of G-spaces is equivalent to the homotopy theory of diagrams of spaces on the orbit category O(G) of G. In the latter homotopy theory, the weak equivalences are "levelwise" as is usual in the homotopy theory of diagrams.
I'm less sure about the (∞,1)-categorical versions, but I would expect that the (∞,1)-category associated to a levelwise model structure on O(G)-diagrams will be essentially the (∞,1)-category of functors from O(G) to the (∞,1)-category of spaces. That ought to imply that it is locally presentable as well.
One might guess that the equivariant stable homotopy category would be the "stabilization" of this (∞,1)-category, but that's not entirely obvious to me. The point at issue is that there are two kinds of G-spectra: "naive" G-spectra, which are indexed on integers, and "true" G-spectra, which are indexed on G-representations. It seems possible to me that the standard "stabilization" process of an (∞,1)-category will only stabilize with respect to integers.