### Equivariant Stable Homotopy Theory

#### Posted by Mike Shulman

Over the past week (at least), Urs and I and some others have been trying to understand *equivariant stable homotopy theory* from a higher-categorical point of view, with some help from experts like Peter May, John Greenlees, and Megan Shulman. Unfortunately, the online part of the discussion has been taking place simultaneously in two different, hard-to-find, and arguably inappropriate places, namely the nForum and the comments on a mostly unrelated thread. So I thought I would give it a thread of its own, along with with an easier-to-read introduction and summary to bring new people into the discussion.

Before we dive in, let me say that equivariant things can be tricky and involve lots of traps for the categorically-minded. (At least, there are a lot of things that trapped me.) If you need some motivation to spur you on and keep you going, note that the recent proof of the Kervaire invariant one problem involved the tools of equivariant homotopy theory in very important ways. So this stuff really is good for something.

First let’s ignore the word “stable” and ask, what is equivariant homotopy theory? Algebraic topologists who study it will tell you that it’s like ordinary homotopy theory except that there’s a group $G$ acting on everything in sight. In complete generality, $G$ could be an arbitrary topological group, but usually they take it to be either a discrete group, a compact Lie group, or (the intersection of the two) a finite discrete group.

However, I think this perspective is misleading to a higher category theorist. (At least, it was misleading to me at first.) What immediately jumps to my mind when I hear “homotopy theory of spaces with $G$-actions” is the $(\infty,1)$-category of $\infty$-groupoids with a $G$-action, i.e. the $(\infty,1)$-functor category $[B G, \infty Gprd]$. That $(\infty,1)$-category ought to be presented by a model category of $G$-spaces in which the weak equivalences, like those in most model categories of diagrams, are objectwise. However, those are *not* the weak equivalences in the model category of $G$-spaces that equivariant algebraic topologists are interested in. Rather, their weak equivalences are the maps $f\colon X\to Y$ of $G$-spaces which induce weak equivalences $f^H\colon X^H \to Y^H$ on spaces of $H$-fixed points, for all (closed) subgroups $H\le G$.

Looking at that, you might guess that from a higher-categorical perspective, what we’re looking at is instead a theory of certain *diagrams* of $\infty$-groupoids, one for each $H\le G$. And in fact, that’s true: if $\mathcal{O}_G$ denotes the orbit category of $G$, then the above model structure on $G$-spaces is equivalent to the ordinary diagram-category model structure (with objectwise weak equivalences) on $[\mathcal{O}_G^{op},Top]$. This is called *Elmendorf’s theorem* (for the obvious reason). So from a higher-categorical perspective, we might try to think of equivariant algebraic topology as the study of presheaves of $\infty$-groupoids on $\mathcal{O}_G$.

It’s not yet clear to me whether *all* aspects of “classical” equivariant homotopy theory are accessible from this higher-categorical viewpoint; part of the goal of this discussion is to figure that out.

One of the main things we’ve been thinking about is cohomology and *stable* equivariant homotopy theory, i.e. the equivariant analogue of spectra. Here I think I will just refer new readers to this comment, which explains the sort of cohomology and spectra that come up. In particular, there are two kinds of $G$-spectra (well, actually, a whole slew of different kinds, with the most interesting being the two extremes). “Naive” $G$-spectra are what the naive $(\infty,1)$-category-theorist would initially write down, namely objects of the stabilization of the $(\infty,1)$-category of “$G$-spaces” considered above (remember that means to consider all the fixed-point spaces specially). On the other hand we have “genuine” $G$-spectra, built out of a set of spaces indexed not just by integers, but by all finite-dimensional representations of $G$. Just as nonequivariant spectra represent (generalized) cohomology theories, a naive $G$-spectrum represents a Bredon cohomology theory, and a genuine one represents an $RO(G)$-graded Bredon cohomology theory.

Now there are lots of questions left! Here are some.

- Is the stable $(\infty,1)$-category of genuine $G$-spectra naturally presented as the stabilization of something?

The stable Giraud theorem seems like it might be relevant to this, and likewise the Schwede-Shipley classification of stable model categories, but the two aren’t quite the same. Lurie’s stable Giraud theorem says that any stable $(\infty,1)$-category is equivalent to a left-exact localization of the stabilization of an $(\infty,1)$-category of presheaves (on some domain $(\infty,1)$-category), while the Schwede-Shipley theorem is that any stable model category is equivalent to a model category of diagrams of spectra on some spectrally enriched category. We’ve been discussing this a bit starting here, and we can continue below.

- Is there some “canonical” way to guess the right notion of “grading” for genuine spectra? For instance, if we start with an arbitrary $(\infty,1)$-topos in place of $G$-spaces, would there be a corresponding notion of “$RO(G)$” and “genuine spectrum” generalizing the “naive spectra” in the naive stabilization?

Urs has some suggestions about this here which I have not digested yet.

- Is there an equivariant analogue of infinite loop space machines? That is, can we define a notion of “grouplike $E_\infty$ $G$-space” which can be delooped into a (genuine) $G$-spectrum?

Peter May says yes, at least when $G$ is finite. But I’m still trying to understand intuitively why his definition of “$E_\infty$ $G$-operad” is correct.

## Re: Equivariant Stable Homotopy Theory

Thanks, Mike, good idea. I was feeling bad for highjacking David Corfield’s thread, too.

Or better yet, the $n$Lab entry:

This contains your comment in full beauty in the section Bredon equivariant cohomology, but also some other related stuff.

There is also

that features an explicit statement of

Elmendorf’s theoremwith references .And similarly there is

with some basic explicit definitions. (Though everything pretty stubby, still.)

The Schwede-Schipley theorem is (in a second) at

I’ll have to call it quits for today soon, but am looking forward to following up on this later.