(∞, 1)-categorical description of equivariant homotopy theory - MathOverflow

(∞, 1)-categorical description of equivariant homotopy theory

2009-10-28T22:28:24Z

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.)

Answer by Mike Shulman for (∞, 1)-categorical description of equivariant homotopy theory

2009-10-28T22:58:01Z

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.

Answer by David Treumann for (∞, 1)-categorical description of equivariant homotopy theory

2009-10-28T22:58:31Z

Let C be the category of homogeneous G-manifolds; the hom sets have a natural topology so you can consider C as an infinity-category. The equivariant homotopy category is the category of contravariant functors from C to the infinity-category of spaces. You build such a functor out of an honest G-space by restricting Hom_G(-,X) to C.

I think this answer is sort of disappointing: it says that all that algebraic topology can see in a G-space are the fixed point sets with respect to subgroups. What are the theorems along these lines that justify this definition?

According to the discussion below, the answer is Whitehead's theorem: any weak G-homotopy equivalence between tame enough G-spaces--at least, all G-CW complexes (Whitehead) and all smooth G-manifolds (Illman)--is a strong G-homotopy equivalence. "Weak" means that the map induces an isomorphism on homotopy groups of all fixed-point sets, and "strong" means that there's an equivariant map backwards so that the compositions are equivariantly homotopic to the identity maps.

Answer by Urs Schreiber for (∞, 1)-categorical description of equivariant homotopy theory

2009-10-29T00:17:25Z

I would say that

a) cohomology is in any case something defined in some (oo,1)-topos (maybe secretly so, but still) -- details and further links on this point of view are at nLab:cohomology

b) from that general point of view there is a very general definition of equivariant cohomology, as indicated at nLab:equivariant cohomology

More precisely, this is (the generalization of) Borel equivariant cohomology . See the remark at A Survey of Elliptic Cohomology: equivariant cohomology - Borel equivariant cohomology.

Answer by Mark Hovey for (∞, 1)-categorical description of equivariant homotopy theory

2009-11-17T18:13:02Z

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.