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 S^n-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.)