Are $G$-spectra the same as modules over a "group ring spectrum"?

Question

Let $G$ be a finite group (maybe this will also work when $G$ is compact, or something, but to be safe we'll let it be finite). I imagine it's quite natural to ask: is the category of $G$-spectra equivalent to the category of module spectra over some ring spectrum, probably denoted by $SG$?

For definiteness, we can take "the" category of $G$-spectra to be the symmetric-spectra model (or orthogonal if we want to try and deal with compact groups), and similarly for modules over a ring spectrum.

I imagine if such a thing existed it would have the property that $\pi_0SG$ should be the actual group ring, $SG$ should be an $A_\infty$-ring, and if $G$ is abelian it should be an $E_\infty$-ring.

It seems like such a thing should exist since the ($\infty$-)category of $G$-spectra (at least coming from, say, the orthogonal model) looks presentable and generated by the equivariant sphere... so by some theorem in Lurie it should be equivalent to an ($\infty$ -)category of modules over some ring spectrum.

A brief search online and glance at May's book on the subject revealed nothing, but I could have easily missed it. Any pointers to the literature or brief epositions of a construction of such a thing would be much appreciated!

(I would say "take the free spectrum on the objects of $G$ and mod out by some relations" and maybe this is how it's done, but I wonder how to make this precise in the "brave new" world, also such a construction probably would not generalize to the compact group case, so maybe there's some better way of doing it.)

Answer 1

Eric wrote a really nice response telling that your initial hope is incorrect and why. I'd just like to write some positive results that you can find.

Disclaimer: I understand little to nothing about the case of a compact Lie group.

Schwede and Shipley have a paper entitled "Stable model categories are categories of modules" from 2003. In particular, G-spectra form a stable model category to which their results apply. Schwede-Shipley show that if you pick a set of "generators", then you'll get an category $I$ enriched in spectra, with $Hom_I(i,j)$ being a spectrum together with units $\mathbb S \to Hom_I(i,i)$ and composition maps $$Hom_I(j,k) \wedge Hom_I(i,j) \to Hom_I(i,k)$$ which are unital and associative. (This is the spectrum version of a DG-category, if you like). Then there is an equivalence between $G$-spectra and enriched functors from $I$ to the category of spectra.

In $G$-spectra, we can pick generators given by the spectra $\Sigma^\infty G/H_+$, which are representing objects for the standard "fixed point" functors. So a $G$-spectrum is equivalent to the data of a collection of spectra $Y^H$ as $H$ ranges over the subgroups of $G$, together with "action maps" $$F(\Sigma^\infty G/H_+, \Sigma^\infty G/K_+) \wedge Y^H \to Y^K$$ that are unital and associative.

If you're feeling like it, you could instead replace several generators with $\bigvee_H G/H_+$, and establish $G$-spectra as equivalent to modules over one ring spectrum which is a big "matrix algebra" containing a bunch of commuting idempotents. It's not clear to me whether this is generally profitable. (It certainly doesn't make looking at the symmetric monoidal structure on $G$-spectra easier.)

To go further, we need to specify a little about which category of $G$-spectra you're interested in. This is often phrased in terms of a choice of universe.

At one end, you have $G$-spectra indexed on the trivial universe, which are formed by taking the category of $G$-spaces and inverting the suspension functor. There's a "coalescing" result of Elmendorf (his "Systems of fixed point sets") that essentially shows that the homotopy category of $G$-spaces is equivalent to the homotopy category of functors from the orbit category of $G$ to spaces; $G$-spectra indexed on the trivial universe satisfy a similar result.

At the other end, you have $G$-spectra indexed on a complete universe, where all the spheres based on orthogonal representations of $G$ become invertible. These are more complicated, because they're generated by more than just actions $g: Y^H \to Y^{gHg^{-1}}$ and restrictions $Y^K \to Y^H$ for $H < K$. They also have transfer maps.

If you've done any looking into $G$-spectra, you've probably encountered the notion of a Mackey functor, which is a collection of abelian groups with restriction, transfer, and conjugation maps. One compact way to phrase this is that Mackey functors are additive functors from the "Burnside category" to the category of abelian groups. $G$-spectra indexed on a complete universe satisfy a similar result: they are enriched functors from a topological Burnside category to the category of spectra. In particular, every $G$-spectrum produces a Mackey functor to the stable homotopy category. (There are several places I could insert some more or less gratuitous $\infty$-category theory here.) I know that Clark Barwick has given several talks on this, and is likely in the process of writing it up.

Whether Mackey functors make you happy might depend on whether you're in that pleasant zone between understanding their definitions and trying to do serious homological algebra with them. While I'm writing pithy asides, it's kind of depressing that the Burnside category doesn't have entries on Wikipedia or the nLab for me to link to, and Mackey functors only have this. Many of the presentations in the literature are worth looking at.

There are several categories of $G$-spectra in between, and I don't know much about the general properties of those.

Answer 2

The "group ring spectrum" $SG$ you ask for does indeed exist, but modules over it are not the same as $G$-spectra. Indeed, $SG$ is just the suspension spectrum $\Sigma^\infty_+ G$ of $G$ as a discrete space. The suspension spectrum of any $A_\infty$ space is naturally an $A_\infty$ ring spectrum, so the group structure on $G$ gives an $A_\infty$ structure on $SG$. An $SG$-module can be shown to be the same thing as a spectrum with a (coherent) action of $G$.

Any $G$-spectrum has an "underlying" non-equivariant spectrum which has the natural structure of an $SG$-module. However, this is not an equivalence of categories, and the basic reason is that they correspond to different notions of "weak equivalence" of equivariant objects. For simplicity, I'll describe this in the unstable setting of $G$-spaces. If $X$ and $Y$ are two spaces with an action of a group $G$ and $f:X\to Y$ is an equivariant map, there are two things we might mean when we say $f$ is an "equivariant homotopy equivalence". The first is that $f$ is an ordinary homotopy equivalence of the spaces $X$ and $Y$ which happens to also be an equivariant map. The second is that $f$ has a homotopy inverse internal to the category of $G$-spaces: there exists another equivariant map $g:Y\to X$ such that the compositions $fg$ and $gf$ are homotopy to the identity through equivariant maps. This notion is much stronger. For example, the map $EG\to *$ is a homotopy equivalence in the weaker sense, but not in this stronger sense, because $EG$ has no fixed points so there are no equivariant maps $*\to EG$. If we restrict to $G$-CW complexes (a natural equivariant generalization of CW-complexes), it turns out that a map $X\to Y$ is an equivariant equivalence in this stronger sense iff for every subgroup $H\subseteq G$, the induced map $X^H\to Y^H$ on fixed points is a homotopy equivalence.

The category of $SG$-modules is a stable version of $G$-spaces under the first, weaker notion of equivalence. Indeed, as with any ring spectrum, a weak equivalence of $SG$-modules is just a map of $SG$-modules which happens to be a weak equivalence of underlying spectra (though an inverse which is actually an $SG$-module map can be found if we're willing to take cofibrant and fibrant replacements of our modules, which corresponds to replacing a $G$-space $X$ with $EG \times X$). On the other hand, the category of $G$-spectra is a stable version of $G$-spaces under the second, stronger notion of equivalence.