TITLE: (∞, 1)-categorical description of equivariant homotopy theory
QUESTION [24 upvotes]: 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.)

REPLY [12 votes]: I have been trying to hold myself back from answering this, because I am not entirely sure my view on this is accurate.  To me, it seems like a G-spectrum should be a spectrum with an action of G on it, full stop.  Obviously you have to specify your notion of spectrum, and obviously if you want to include topological groups G it had better be a symmetric monoidal category of spectra that is enriched over topological spaces.  So you could take S-modules of EKMM or orthogonal spectra (my personal favorite) or symmetric spectra based on topological spaces.  With any of these categories, there is a notion of a G-spectrum, by which I mean a spectrum with an action of G.  
I can hear you objecting--you must be being too naive--what about complete G-universes?  I take the point of view that picking a universe corresponds to picking a model structure on the one God-given category of G-spectra.  Picking a smaller universe just means localizing the model structure.  So the complete universe is the "initial" one, and every other universe is a localization of the complete universe.  The naive universe is the "terminal" one, in the sense that it is a localization of every other universe.  There are lots of universes corresponding to model structures in between these.  
I cannot now remember how these model structures are supposed to go, but I believe that both Neil Strickland and Tony Elmendorf have separately written something about this approach.  Tony's might be part of a joint paper, I can't remember.  I think it is just a different way to look at things,  but it goes so much against the prevailing viewpoint that it has not gotten so much traction.  
Again, I have to confess that I am working from memory from something I probably did not completely understand.  Possibly Mike Shulman or someone else will be able to convince me I am completely wrong.