## January 24, 2010

### 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?

• 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.

Posted at January 24, 2010 9:17 PM UTC

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### Re: Equivariant Stable Homotopy Theory

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

Here I think I will just refer new readers to this comment,

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 theorem with 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.

Posted by: Urs Schreiber on January 24, 2010 10:30 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Thanks. There are so many links to give that it’s hard to fit them all in. (-:

Posted by: Mike Shulman on January 24, 2010 11:01 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I like that about the Café that a technical discussion can break out in the middle of speculative discussion. In a private e-mail someone observed of the discussion that it “seems to be oscillating between spirituality and higher homotopy theory, which may actually be two names for the same thing.”

But good to have a focused thread here, so let me just ask again, if we really believe in the Baez-Dolan idea that ordinary spectra are a type of $\mathbb{Z}$-groupoid and we now seem to need more exotic gradings by $RO(G)$, does this mean there are $RO(G)$-groupoids?

Is part of the reason for the restriction to $n$-categories that the representations of the trivial group are classified by natural numbers?

Posted by: David Corfield on January 25, 2010 8:39 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I see John and Urs were taking $\mathbb{Z}$-categories seriously back here. Urs even wondered whether we might end up with things like $\mathbb{Z}^{\mathbb{Z}}$-categories.

Posted by: David Corfield on January 25, 2010 9:22 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

let me just ask again, if we really believe in the Baez-Dolan idea that ordinary spectra are a type of $\mathbb{Z}$-groupoid

My best guess for what a (not-necessarily strict) $\mathbb{Z}$-groupoid might actually be, is that it is the structure that is used in the last part of

to discuss spectrum valued $\infty$-sheaves, and which on the $n$Lab we gave the name combinatorial spectrum. This is a $\mathbb{Z}$-graded pointed set that has finite but arbitrarily many pointed face and degenracy maps in each degree, satisfying the usual simplicial identities (it has an infinite number of face and degeneracy maps in each degree of which however only finitely many are allowed not to be constant on the point).

I don’t think I have seen this construction used anywhere apart from this article (but that need not mean much), and I always wondered how it relates to the now standard notion(s) of spectra, but then never really spent serious energy to find out.

But it is a very elegant and nice definition, and certainly Kenneth Brown behaves in this article as if it were clear that this is the notion of spectrum (though I should maybe read the last part again now, maybe he says something about it and I am just making a fool of myself here by saying that there is a mystery).

If this is right that Ken Brown’s combinatorial spectra are equivalent to the ordinary notion of spectra, then I believe thinking of them as $\mathbb{Z}$-groupoids would be entirely justfied and useful, as they directly generalize the notion of Kan complex form $\mathbb{N}$-grading to $\mathbb{Z}$-grading.

In that case, how to obtain similar combinatorial definitions for spectra stabilized with respect to non-standard types of loop space objects, as we are discussing here, I don’t know.

Posted by: Urs Schreiber on January 25, 2010 11:29 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

As mentioned in the query box at combinatorial spectrum, these were originally invented by Kan, who proved that their homotopy category is equivalent to the usual stable homotopy category. The idea is that a combinatorial spectrum is built out of a naive prespectrum of simplicial sets (that is, a sequence of based simplicial sets $X_n$ with maps $\Sigma X_n\to X_{n+1}$) by making the $k$-simplices of $X_n$ into $(k−n)$-simplices in the combinatorial spectrum. I think this was pre-model categories (or at least pre- wide use of model categories), but it seems not too unlikely that there is a model category of combinatorial spectra that is Quillen equivalent to the usual ones.

I agree that these give a very nice picture of spectra as $\mathbb{Z}$-groupoids. But I think the reason no one uses these any more is that no one has managed to come up with a smash product of them that is associative on the point-set level, so none of the modern “brave new algebra” of ring spectra, module spectra, etc. works. I’m not sure whether anyone has really tried, though; I expect that you would probably have to modify the definition to take symmetries into account, e.g. starting with symmetric prespectra instead of ordinary prespectra.

Posted by: Mike Shulman on January 25, 2010 4:08 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

As mentioned in the query box at combinatorial spectrum,

Let’s move that out of the query box! Together with the statement about smash products that you just made. That’s useful information.

Posted by: Urs Schreiber on January 25, 2010 4:24 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

does this mean there are RO(G)-groupoids?

Maybe. Although a genuine G-spectrum doesn’t just have a space for each representation, it has a G-space.

Actually, I’m never quite sure whether spectra should be $\mathbb{Z}$-groupoids or pointed $\mathbb{Z}$-groupoids. Maybe I’m confused.

Posted by: Mike Shulman on January 25, 2010 8:05 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I think it will help not to forget that the “good” notion of spectra relies on a geometry. Stable homotopy theory is made to give sense to cohomology theories with a strong geometric flavour, the very fundamental ones being cobordism and K-theory. These two examples need, as an input, the notion of (equivariant) vector bundle (equivariant vector bundle over the point are the finite dimensional representations of the group). To such a thing, we associate a Thom space, and these are used to produce Thom spectra, which, by definition will involve this “non naive” grading. To make sense of these constructions, we need to be in a world where all the Thom spaces are invertible (not only the circle). So the general data is not only an $(\infty,1)$-topos, but a ringed $(\infty,1)$-topos $T$ (which you may think of a notion of $(\infty,1)$-topos endowed with a notion of vector bundle). We have the same issue with the homotopy of schemes: we start from (Nisnevich) sheaves on the category of smooth schemes over a given base scheme $S$, endowed with the canonical ring $\mathbf{A}^1$ (the affine line). This affine line is used to define homotopies and to define vector bundles, from which we also get algebraic Thom spectra (which we have to invert to get the stable homotopy of schemes). This is where the yoga of weights appear in algebraic geometry.

With this example in mind, it seems to me that equivariant homotopy theory has, as an input, the geometry of $G$-equivariant manifolds (i.e. the category of $G$-equivariant manifolds and the notion of $G$-equivariant vector bundles). The issue of choosing a topology on this site might also be related to the question of “naive” equivariant homotopy theory versus the “genuine” approach. For instance, when I gave the example of the homotopy theory of schemes, I mentionned the Nisnevich topology. But, of course, one might take others, like the étale topology. The interesting thing is that, when the base scheme is the spectrum of the field of real numbers, the real points functor induces a left Quillen functor from the Nisnevich version of the homotopy theory of schemes to the “genuine” $\mathbf{Z}/2\mathbf{Z}$-equivariant homotopy theory, while the same functor induces a left Quillen functor from the étale verson of the homotopy theory of schemes to the “naive” $\mathbf{Z}/2\mathbf{Z}$-equivariant homotopy theory.

So, it seems unnatural to me to wish for a canonical way to obtain a “genuine” grading if you don’t have a minimum of geometry as an input: this would sound like trying to reconstruct canonically a scheme from its underlying topological space (or topos) alone…

Posted by: Denis-Charles Cisinski on January 25, 2010 1:31 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

So the general data is not only an $(\infty,1)$-topos, but a ringed $(\infty,1)$-topos T […] endowed with the canonical ring $\mathbb{A}^1$ (the affine line).

Yes! I think that’s exactly what I have been saying, as Mike mentioned:

I said we want to build the spheres using the given geometric line object (or ring object, if you prefer that).

I like to think of the context of playing all these games with geometric paths and spheres, $\mathbb{A}^1$-homotopies etc. as being any path-structured $(\infty,1)$-topos.

Posted by: Urs Schreiber on January 25, 2010 2:09 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I said we want to build the spheres using the given geometric line object (or ring object, if you prefer that).

I’m working on trying to understand line objects and stuff. I do like the idea that the input should involve not just a topos but a topos with some additional “geometry.” However your suggestion at the other thread doesn’t seem quite right yet to me, because I’m pretty sure that if you just pick some particular interval object $I$ (with of course will have some fixed $G$-action) then the spheres of the form $I^n/\partial I^n$ won’t give you all the representation spheres. But maybe there is some other way of arriving at them. I wish I understood homotopy of schemes better; the suggestion of a relationship there is quite intriguing.

Posted by: Mike Shulman on January 25, 2010 10:28 PM | Permalink | PGP Sig | Reply to this

### Re: Equivariant Stable Homotopy Theory

The interesting thing is that, when the base scheme is the spectrum of the field of real numbers, the real points functor induces a left Quillen functor from the Nisnevich version of the homotopy theory of schemes to the “genuine” $\mathbb{Z}/2\mathbb{Z}$-equivariant homotopy theory, while the same functor induces a left Quillen functor from the étale verson of the homotopy theory of schemes to the “naive” $\mathbb{Z}/2\mathbb{Z}$-equivariant homotopy theory.

I think I see what you mean, but I am not sure I see how this comes about. Do you have a reference for this? Is this in your motivic articles?

Posted by: Urs Schreiber on January 25, 2010 2:17 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

the real points functor induces a left Quillen functor from the Nisnevich version of the homotopy theory of schemes to the “genuine” $\mathbf{Z}/2\mathbf{Z}$-equivariant homotopy theory

Do you really mean the real points functor? I thought it was the complex points that had a $\mathbf{Z}/2\mathbf{Z}$-action coming from complex conjugation.

Posted by: Mike Shulman on January 25, 2010 10:42 PM | Permalink | PGP Sig | Reply to this

### Re: Equivariant Stable Homotopy Theory

I’ve been trying to understand what an equivariant E-infinity operad is too, for a while now. I don’t have a good explanation. But here is an observation, which is perhaps tangentially relevant.

* We usually use E-infinity operads as a means to talk about E-infinity algebras, i.e., gadgets which are “commutative monoid” type objects (at least in some homotopy sense).

Of course, you can’t talk about commutative monoids (in any sense) unless you have a symmetric monoidal category around. And E-infinity operads play a role here too: a symmetric monoidal (infinity,1)-category can be modelled as an (infinity,1)-category equipped with an action by an E-infinity operad. (I’ll assert this as though it’s obvious; I’m unaware that anyone has treated symmetric monoidal (infinity,1)-categories this way, though I think Lurie has modelled them using Gamma-spaces, which is pretty close.)

* What about G-spaces? I’m interested in the homotopy theory of G-spaces, so I’m happy to use the Elmendorff model: presheaves of spaces on the orbit category of G. This gives us a fine (infinity,1)-category, the “(infinity,1)-category of G-spaces” I’ll call it. This thing is symmetric monoidal, using product.

* But I claim there’s something more: G-spaces don’t just form an (infinity,1)-category. They form an “(infinty,1)-category internal to G-spaces”. (A vanilla (infinity,1)-category is just a one internal to spaces.)

(Of course, there is an issue of universes here, which I’ll blithely ignore!)

How does this work? I’ll give you a presheaf of (infinity,1)-categories on the orbit category of G. This assigns to a G-orbit X the (infinity,1)-category “F(X) := (G-spaces)/X”, i.e., the slice (infinity,1)-category of G-spaces over X. Given a map X –> Y, the functor F(X) –> F(Y) is the one induced by pulling back along the map.

(This construction clearly works for any category of presheaves of spaces; the (infinity,1)-category of presheaves of spaces on C is naturally *internal* in presheaves on C. In fact, (infinity,1)-toposes are always internal to themselves.)

Note that if X=G/H, then F(X) = (G-spaces)/X is equivalent to the (infinity,1)-category of H-spaces. So you can think of E as sending G/H |–> (H-spaces).

* So G-spaces form an (infinity,1)-category in G-spaces, which I’ll call F. But we also have a fancy G-equivariant operad in G-spaces. Could it be that F is an algebra over the G-equivariant operad? This would somehow mean that G-spaces are “even more” than merely symmetric monoidal, but have some kind of “equivariant symmetric monoidalness”.

I’m very sure the answer is yes. To say why would mean talking a lot about the equivariant E-infinity operad, and I’m running of out steam here. But what will happen is that the “extra” monoidalness encodes “norm constructions”. For instance, given a space X, you can (for a *finite* group G) form a G-space N^G(X), whose underlying space looks like the set of functions G –> X (and thus, like a G-indexed product of copies of X), but where the G-action comes from permuting the factors. (And more generally, given an H-space X you can form N^G_H(X).)

(A spectrum version of this norm construction shows up in the solution to the Kervaire invariant.)

Posted by: Charles Rezk on January 25, 2010 2:30 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

And $E_\infty$ operads play a role here too: a symmetric monoidal $(\infty,1)$-category can be modelled as an $(\infty,1)$-category equipped with an action by an $E_\infty$ operad. (I’ll assert this as though it’s obvious; I’m unaware that anyone has treated symmetric monoidal $(\infty,1)$-categories this way, though I think Lurie has modelled them using Gamma-spaces, which is pretty close.)

In the updated version of Jacob Lurie’s Commutative Algebra it is done this way. More developments along these lines, such as the stabilization hypothesis in these terms, is in $\mathbb{E}_k$-Algebras.

And if I recall correctly what David Ben-Zvi recounted here once, this update goes back to or was triggered by the PhD thesis of John Francis, who went ahead and defined $k$-fold monoidal $(\infty,1)$-categories as algebras in $(\infty,1)$Cat over $E_k$-operads.

My impression is this event has pushed the announced Spectral Schemes from position 6 further off, with position 6 now being taken by $\mathbb{E}_k$-Algebras . But that’s just me guessing from what I can see around me.

Concerning what $E_\infty$-operads with respect to a given $\infty$-stack $(\infty,1)$-topos are, here is, for the record, my running hypothesis which started this entire discussion with Mike in the first place:

Hypothesis. An $E_\infty$-operad in an $(\infty,1)$-topos of $\infty$-stacks on an $(\infty,1)$-site $C$ is the object in the $(\infty,1)$-category of “dendroidal stacks” on $C$ represented by the objectwise $E_\infty$(Top)-valued dendroidal stack.

More concretely, in model category theoretic terms: let $SSet Cat[C^{op}, dSet]_{proj}$ be the $SSet_{Joyal}$-enriched projective global model category of $SSet$-enriched presheaves with values in dendroidal sets with “the” model structure on dendroidal sets, and let $SSet Cat[C^{op}, dSet]_{proj}^{loc}$ be its left Bousfield localization, using Barwick’s theorem, at the set of all Čech nerve projections for covering families.

Then the $N_d(E_k(Top))$-valued presheaf in there, fibrant-cofibrantly replaced, is the $E_k(Sh_{(\infty,1)}(C))$-operad.

Well, maybe it’s less a hypothesis than a motivating question. But I think it should be true that in a lined $(\infty,1)$-sheaf $(\infty,1)$-topos with line object $I$, the geometric $k$-fold look space objects $[I^k/\partial I^k, X]$ have an action not (just) of an $E_k(Top)$-operad, but of a dendroidal sheaf, along such lines.

This discussion got started when a comment by John Greenlees made me wonder if stable equivariant homotopy theory might be a comparatively simple (hah :-) testing ground for this, due to its comparatively simple underlying site structure.

Posted by: Urs Schreiber on January 25, 2010 3:44 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Urs, what is $N_d(E_k(Top))$ suppose to represent?

Posted by: Charles Rezk on January 25, 2010 6:43 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Urs, what is $N_d(E_k(Top))$ suppose to represent?

Sorry, I should have been more explicit:

the dendroidal nerve of an ordinary topological $E_k$-operad. So just the $E_k$-operad in its incarnation as a quasi-operad.

Posted by: Urs Schreiber on January 25, 2010 8:09 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

With the equivariant homotopy category understood as the homotopy category of the $I$-homotopy localizaton of $\infty$-sheaves on $G Top$, I now finally give the promised details on my idea for the $G$-equivariant little $k$-cubes operad that I was talking about:

Let $C := sPSh(G Top)$ be the category of simplicial presheaves on $G Top$ and let $I = [0,1]$ be the standard interval object in there (with trivial $G$-action).

As described at interval object: little 1-cubes space, for every object $X \in C$ this induces a planar dendroidal object $Paths X : \Omega_{p}^{op} \to C$ whose object in $C$ over a given tree is $[I^{\vee n} , X]$ for $n$ the number of leaves of the planar tree.

In particular there is $Paths I$, which is a model for the little 1-cubes operad in $C$. The action of $Paths I$ on $Paths X$ is exibited by a map $Paths X \to Paths I$, which should be the map classified by the corresponding action $Paths I \to End [I,X]$.

For $X$ pointed write $Loops_* X \subset Paths X$ for the dendroidal subobject of paths with endpoints sitting at $*$.

We may then extend these planar dendroidal objects to dendroidal objects along the canonical functor $\Omega_{p} \to \Omega$, thus obtaining the corresponding (symmetric-)operad structures.

The dendroidal objects in $[\Omega^{op}, sPSh(G Top)]$ are equivalently objects in $PSh(G Top, dSet)$, which I regard equipped with the projective model structure of $dSet$-valued presheaves, using the standard model structure on dendroidal sets. The image of this under first Cech localization and then $I$-homotopy localization

$PSh(G Top, dSet) \stackrel{left\;Bousf.\; loc}{\to} PSh(G Top, dSet)_{loc}^I \to Ho(PSh(G Top, dSet)_{loc}^I) \simeq G Top_{loc}$

$dendroidal-homotopy-stackify : Loops_* X \mapsto \bar{(Loops_* X)}$

(which is of course the identity map on the categories underlying the model categories) I am thinking could be a good model for the $G$-equivariant action $E_1$-operad of loops in $X$.

An entirely analogous discussion should go through for any $k \leq \infty$, leading to $G$-equivariant $E_k$-action operads on $k$-fold geometric loop space objects.

That’s the idea, anyway.

Posted by: Urs Schreiber on January 30, 2010 12:13 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Reading over this, I see that I didn’t complete my thought.

G-equivariant stable homotopy theory (both “naive” and “non-naive”) are symmetric monoidal, under smash product. This structure is compatable with the unstable symmetric monoidal structure: the suspension spectrum functor is monoidal.

Question: Is equivariant stable homotopy theory “fancy equivariant symmetric monoidal”?

If yes, then equivariant stable homotopy theory should have a norm construction, compatible with the space level norm.

For instance, let G be a finite group. Then if R denotes the line (viewed as a non-equivariant space), then N^G(R), as a G-space, is the real regular G representation V.

Since the spectrum level norm should be compatible, this means that if S^1 is the suspension spectrum of the one point compactification of R, then N^G(S^1) should be the suspension spectrum of the one-point compactification of V; i.e., N^G(S^1) = S^V.

Now, the norm construction is monoidal: on spectra, N^G should take smash products of spectra to smash products of G-spectra. But when we form a stable category, we have to “formally invert” S^1. So what we discover (I think), is that if you want to have a “fancy equivariant symmetric monodial” category of G-spectra, you have to “invert” its norm as well.

In other words, I’d expect that: (i) naive equivariant stable homotopy isn’t fancy equivariant monoidal, (ii) non-naive equivariant stable homotopy is fancy equivariant monoidal. Furthermore, I’d conjecture that non-naive G-equivariant stable homotopy theory is the universal stabilization of unstable G-equivariant homotopy theory which is fancy monoidal.

(This is still a hazy idea for me, I’m entirely sure how to formalize it.)

Posted by: Charles Rezk on January 25, 2010 6:36 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Charles Rezk explained:

$G$-spaces don’t just form an $(\infty,1)$-category. They form an “$(\infty,1)$-category internal to $G$-spaces”. (A vanilla $(\infty,1)$-category is just a one internal to spaces.)

[…]

How does this work? I’ll give you a presheaf of $(\infinity,1)$-categories on the orbit category of $G$. This assigns to a $G$-orbit $X$ the $(\infty,1)$-category “$F(X) := (G-spaces)/X$”, i.e., the slice $(\infty,1)$-category of $G$-spaces over $X$. […]

Okay, I am following. These presheaves of over-$(\infty,1)$-categories generally seem to be a pretty fundamental thing. When objectwise stabilized, it seems one should/ or may think of them as the “quasicoherent $\infty$-stack of modules” canonically determined by the given context. (Along the lines we recalled at here). I was wondering what I should think of them before stabilization. It seems you are giving one perspective here on that.

But we also have a fancy $G$-equivariant operad in $G$-spaces. Could it be that $F$ is an algebra over the $G$-equivariant operad? […] I’m very sure the answer is yes. To say why would mean talking a lot about the equivariant $E_\infty$ operad, and I’m running of out steam here.

Maybe when you have regained some steam, you can talk about this a bit more? I’d be interested in what you have to say here.

I’d conjecture that non-naive $G$-equivariant stable homotopy theory is the universal stabilization of unstable G-equivariant homotopy theory which is fancy monoidal.

Posted by: Urs Schreiber on January 25, 2010 9:03 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

In fact, $(\infty,1)$-toposes are always internal to themselves.

Is that by any chance a different way of talking about the codomain fibration, a.k.a. the self-indexing, which is the canonical way of talking about a topos $E$ as “a category (= fibration = indexed category) in the world of $E$”?

To say why would mean talking a lot about the equivariant $E_\infty$ operad

Can you at least say what you mean by “the” equivariant $E_\infty$ operad? To use a definite article you apparently have a particular one in mind.

Posted by: Mike Shulman on January 25, 2010 10:38 PM | Permalink | PGP Sig | Reply to this

### Re: Equivariant Stable Homotopy Theory

Oops! In my earlier posts, I sometimes wrote “equivariant operad” or “G-equivariant operad”, but I really always meant “E-infinity G-equivariant operad”; that is, the gadget that Peter May introduced for understanding equivariant infinite loop spaces.

Maybe I’m talking about the “codomain fibration”; I’m not sure, it’s a language I don’t use, so I dunno. The assertion that “(infinity,1)-toposes are internal to themselves” is pretty strong. It is *not* true that classical Grothendieck toposes are internal to themselves, in the sense I have in mind; so I can’t show you a “classical” analogue.

Here’s the precise claim about internality. Let E be an (infinity,1)-topos. Let F be the functor E^op –> (infinity,1)-cat defined on objects by

F(X) := ( the slice category E/X ),

and on morphisms by “pullback in E”. Then F converts homotopy colimits into homotopy limits. (This is roughly a reformulation of what Lurie calls the “descent” property of an (infinity,1)-topos.)

Posted by: Charles Rezk on January 25, 2010 11:17 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Then F converts homotopy colimits into homotopy limits. (This is roughly a reformulation of what Lurie calls the “descent” property of an (infinity,1)-topos.)

Classical Grothendieck toposes do satisfy their own version of descent, of course. The corresponding statement is that if $E$ is a Grothendieck topos, then the similarly defined functor $E^{op} \to Cat$ is a stack for the canonical topology. This is a statement about taking certain (2-)colimits in $E$ (regarded as a locally discrete 2-category) to (2-)limits in $Cat$. It doesn’t take all colimits in $E$ to limits in $Cat$; I expect that things are more convenient in the $\infty$-world because $\infty+1=\infty$. But I’m pretty sure this is the classical analogue.

Posted by: Mike Shulman on January 26, 2010 12:07 AM | Permalink | PGP Sig | Reply to this

### Re: Equivariant Stable Homotopy Theory

Yes, I agree with that. But I wouldn’t quite call it being “internal”. If E is a classical topos, then E isn’t a category object of E (or a bigger universe version of E), but rather a category object in some larger beast (stacks on E).

Posted by: Charles Rezk on January 26, 2010 5:39 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

If E is a classical topos, then E isn’t a category object of E (or a bigger universe version of E), but rather a category object in some larger beast (stacks on E).

Well, a stack is already a category; you don’t need to talk about category objects in stacks. (Unless by “stack” you prefer to mean “stack of groupoids”, in which case maybe I should say “2-sheaf” instead of “stack”.) And while it’s true that a stack is not the same as a category object in sheaves, as I mentioned down here every stack can be strictified into an equivalent one which is a category object in sheaves. One place this is proven is in Steve Awodey’s thesis, chapter V.

Granted, this isn’t as convenient or natural as not having to strictify. But I think it’s fair to say, based on this, that the $(\infty,1)$-situation is not conceptually different, just more convenient and natural.

Posted by: Mike Shulman on January 27, 2010 2:03 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Well, I think one problem is that I shouldn’t have chosen the word “internal”.
I really have something very different in mind. Namely, that $(\infty,1)$-toposes have (LARGE) “object classifiers”.

That is, if $E$ is an $(\infty,1)$-topos, there is an OBJECT $\Omega$ in $E$, such that (homotopy classes of) maps $X\to \Omega$ correspond to (isomorphism classes of) objects in the slice category $E/X$. Note that $\Omega$ is characterized up to weak equivalence by this property; it’s a completely natural thing.

I want to think of $\Omega$ as part of the information which describes an internal model of $E$ inside itself. (in particular, it’s the part that describes the objects of $E$.)

An (classical) $1$-topos $E$ doesn’t have an object classifier. It is only allowed to have a subobject classifier, i.e., something that classifies objects in $E/X$ which are monomorphisms. Of course, there is something that you can talk about that does the job of classifying objects of $E$ (a stack!). But such a stack is not an object of $E$; at best, it’s an object of a certain $2$-topos which contains $E$.

Maybe this isn’t really a big deal. But to me, it feels like a big qualitative difference between $1$-toposes and $\infty$-toposes.

Posted by: Charles Rezk on January 27, 2010 5:23 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Hmm. Ignoring size issues so blatantly is difficult for me to swallow. The object classifier of an $(\infty,1)$-topos $E$ is not an object of $E$ itself, it is an object of some version $E'$ of $E$ in a higher universe. That means it isn’t determined up to equivalence by $E$ alone, but depends also on the choice of universe boundary.

Alternately, you can think of the object classifier as an object of $E$, but which only classifies the class of “small” objects in $E$. But the subobject classifier of a 1-topos can also be thought of as classifying the “small” objects in $E$, where now “small” means relative to the regular cardinal $2$. Many 1-toposes also have object classifiers that classify some larger class of “small objects”, such as are studied in algebraic set theory. Of course these classifiers are not objects of the topos itself, but rather internal categories in it; even the subobject classifier is an internal poset. If you only want to classify the core of the slice category, then you can get away with internal groupoids, and since the core of a poset is discrete, the subobject classifier can classify isomorphism classes of subobjects on the nose without being considered as an internal groupoid.

The difference in the $\infty$-case is that any internal groupoid in an $(\infty,1)$-category is representable by an ordinary object, since $\infty+1=\infty$. (If we go all the way to $(\infty,\infty)$-toposes, whatever those might be, then any internal category might to be representable by an ordinary object, so that “object classifiers” could classify the whole slice category, not just its core.) So this is certainly a qualitative difference, and I agree that it is important and useful. I’m just saying that there are analogues of it in the 1-topos world, even if they aren’t as nice.

Posted by: Mike Shulman on January 27, 2010 3:32 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Ignoring size issues so blatantly is difficult for me to swallow. The object classifier of an (∞,1)-topos $E$ is not an object of $E$ itself,

In TheBook the size issues are carefully taken care of: there is for each regular cardinal a classifying object for all “$\kappa$-compact morphisms”.

This is indicated and referenced at object classifier.

Maybe that helps here?

Posted by: Urs Schreiber on January 27, 2010 5:02 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

In TheBook the size issues are carefully taken care of: there is for each regular cardinal a classifying object for all “κ-compact morphisms”.

Yes, that’s basically what I meant by

you can think of the object classifier as an object of E, but which only classifies the class of “small” objects in E.

Posted by: Mike Shulman on January 28, 2010 12:12 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Charles,

Here is how to make your formulas come out pretty printed:

when composing a comment, there is a little pulldown menu on top of your edit pane, that says “text filter”. Choose the text filter called

Markdown with itex to MathML .

When you do that, then everything you type inside dollar signs as in a LaTeX document will be displayed as math, as in a LateX document.

That’s what the “instiki to MathML” takes care of. The “Markdown” part takes care that things included inside stars come out boldface, things inside underscores come out italicized, and lines preceded by a greater-than sign come out as indented quotes.

More on $G$-Spaces from me tomorrow. Have to catch some sleep now…

Posted by: Urs Schreiber on January 26, 2010 1:58 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I remarked cryptically:

Is that by any chance a different way of talking about the codomain fibration, a.k.a. the self-indexing

To expand on that a bit for those listening who may not be familiar with it. For an ordinary 1-topos $S$, if you want to do mathematics “in the world of $S$,” then you can do “small mathematics,” working with individual objects like groups and rings, just internal to $S$, but if you want to work with large categories (say, the category of rings in the world of $S$) you have an issue. Of course you can talk about the category of internal ring objects in $S$, but when doing category theory we also need to talk about set-indexed families of objects of categories: for instance when we say that a category is cocomplete we mean that any set-indexed family of objects has a coproduct. But in the world of $S$, the objects of $S$ play the role of “sets,” so we need a notion of “a family of ring objects indexed by an object $X\in S$”. The obvious notion is an internal ring object in $S/X$.

In general, for every naturally defined large category in the world of $S$ there are corresponding categories defined in each slice categories of $S$, and via pullback these categories fit together into a pseudofunctor $S^{op}\to CAT$ (aka an “indexed category”), or equivalently a fibration over $S$. Such a pseudofunctor/fibration $C$ comes with a notion of “$X$-indexed object of $C$” for each $X\in S$, namely the category $C(X)$, or equivalently the fiber of the fibration over $X$. You can then “do category theory” with fibrations/pseudofunctors in the world of $S$ just as you do with ordinary large categories. The fibration representing $S$ itself, i.e. “the category of sets in the world of $S$,” is the codomain fibration, or the pseudofunctor sending $X$ to the slice category $S/X$ (and maps to pullback functors).

Of course, if you want to work with small categories, then you have the other option of just working with internal categories in $S$. In some ways this is “easier,” e.g. you can use the internal logic and talk as if the objects of $S$ were sets. (The 2-internal logic of the category of fibrations aims to make this possible for fibrations as well.) Every internal category represents an indexed category in the same way that every object represents a presheaf, so “small categories give rise to large categories.”

And if you don’t like working with fibrations at all, i.e. with large categories as large categories, you can also do some universe trickery to make them into small ones. For instance, if your topos $S$ is the category of sheaves on some site $D$, then you can consider the topos $S'$ of sheaves on the same site but with values in larger sets (i.e. you assume a universe, so that $S$ be the category of small-set-valued sheaves, and $S'$ the category of large-set-valued sheaves). Then internal categories in $S'$ can be treated as “large categories in the world of $S$.” For instance, I think this is what Street did in his paper “The petit topos of globular sets.” An indexed category, i.e. pseudofunctor $S^{op}\to CAT$ can be turned into an internal category in $S'$ by first restricting it to a pseudofunctor $D^{op}\to CAT$, then “strictifying” it to a strict functor $D^{op}\to CAT$, which is the same as an internal category in the category of functors $D^{op}\to SET$; which in most cases will be objectwise a sheaf, hence an internal category in $S'$.

So when Charles said:

I’ll give you a presheaf of $(\infty,1)$-categories on the orbit category of $G$. This assigns to a $G$-orbit $X$ the $(\infty,1)$-category “$F(X) \coloneqq (G-spaces)/X$”, i.e., the slice $(\infty,1)$-category of $G$-spaces over $X$. Given a map $X \to Y$, the functor $F(X) \to F(Y)$ is the one induced by pulling back along the map.

It sounds very much to me like the above construction: start with the self-indexing $X\mapsto (G-spaces)/X$ of the category $S = G-spaces$, restrict it to the site, i.e. the orbit category $D = \mathcal{O}_G$, then regard that as an internal category in the category $G-SPACES$ one universe higher up.

Posted by: Mike Shulman on January 25, 2010 11:01 PM | Permalink | PGP Sig | Reply to this

### Re: Equivariant Stable Homotopy Theory

I want to try to describe the way in which the $(\infty,1)$-category of $G$-spaces is supposed to be monoidal over the “$G$-equivariant $E_\infty$ operad”. I learned about equivariant operads from Rekha Santhanam, and much of my understanding of how this monoidal structure works comes out of conversations with David Gepner.

In what follows, “$G$-spaces” always means the $(\infty,1)$-category (=homotopy theory) of $G$-spaces, in the strong sense; that is, weak equivalences are maps which induces weak equivalences on all fixed point spaces.

Fix a finite group $G$. A “fancy G-equivariant $E_\infty$ operad” $U$ is an operad in the symmetric monoidal category of $G$-spaces (under product), characterized by the property: for each $n$ and each subgroup $K\subseteq G\times \Sigma_n$, $U(n)^{K}$ is either (i) contractible, or (ii) empty, depending on whether (i) $K\cap (\{e\}\times \Sigma_n)$ is the trivial subgroup of $G\times \Sigma_n$ or (ii) isn’t. It’s probably better to think of the case (i) subgroups as being the ones which look like graphs of homomorphisms $H\to \Sigma_n$, where $H\subseteq G$ is a subgroup.

(A side remark: although $U$ is an operad from the point of view of category theory, it isn’t really properly an operad in the $(\infty,1)$-category theory sense. The object $U(n)$ is not merely a “an object of $G$-spaces equipped with an action of $\Sigma_n$”, as you would expect, but rather a more sophisticated thing, “an object of $G\times \Sigma_n$-spaces”.)

I want to use Elmendorf’s theorem freely when talking about $G$-spaces. So if I have a $G$-space $X$, I get a presheaf on the orbit category $O=O_G$ of $G$, by $G/H \mapsto X^H$. Conversely, I can reconstruct $X$ (up to $G$-equivariant weak equivalence) from its associated presheaf.

The notable feature of $U(n)$ is that the $H$-fixed points of the quotient space $U(n)/\Sigma_n$ is the classifying space of the category of functors $H\to \Sigma_n$ (for subgroups $H$ of $G$).

I need to think about a generalization of this. If $X$ is a $G$-space, consider $(U(n)\times X^n)/\Sigma_n$. The $H$-fixed points of this map to the $H$-fixed points of $U(n)/\Sigma_n$, so I get a “bundle”

(1)$[(U(n)\times X^n)/\Sigma_n]^H \to (U(n)/\Sigma_n)^H.$

A point in the “base” represents a functor $\rho\colon H\to \Sigma_n$. The “fiber” over $\rho$ is a space $\mathrm{Hom}_G( S_\rho, X)$ of $G$-equivariant maps. What’s $S_\rho$? That’s a $G$-space which sits in a fibration $S_\rho\to G/H$, with fiber equal to $\{1,\dots,n\}$, and with $G$ action determined by the homomorphism $\rho \colon H\to\Sigma_n$. It’s “determined” in the following way: the group $H$ has to act on the fiber over the coset $eH$, and it acts according to the homomorphism $\rho$.

Some examples: If $\rho(H)=\{e\}$, then $S_\rho = \{1,\dots,n\}\times G/H$, and so $\mathrm{Hom}_G(S_\rho, X)= (X^H)^n$. If $\rho\colon H\to \Sigma_n$ describes the left action of $H$ on itself, then $S_\rho= G$, and so $\mathrm{Hom}_G(S_\rho, X)=X$.

Now let $X$ be an algebra for $U$. The data of such an algebra are maps $(U(n)\times X^n)/\Sigma_n \to X$ of $G$-spaces. I want to understand these in terms of fixed points: $[(U(n)\times X^n)/\Sigma_n]^H \to X^H$.

I’ll continue in another post …

Posted by: Charles Rezk on January 26, 2010 7:18 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

… and here’s the continuation.

Given a $G$-space $X$, let $E(X)$ denote the $(\infty,1)$-category of $G$-spaces over $X$. It’s easiest for me to think of $E(X)$ as being a complete Segal space; so $E(X)$ is a simplicial space, and its bottom layer $E_0(X)$ is a space which corresponds to the $(\infty,1)$-groupoid of $G$-spaces over $X$. Note that $E$ is contravariant in $X$; the functor $E(Y)\to E(X)$ is the one induced by pulling back along $X\to Y$.

Since $G$-spaces are an $\infty$-topos, we have descent: in particular, for any two $G$-spaces $X$ and $Y$ the map $E(X\amalg Y)\to E(X)\times E(Y)$ is an equivalence. More generally, the value of $E$ at an arbitrary $G$-space $X$ can be recovered from its values at $G$-orbits.

Note further that $E(G/H)$ is equivalent to the $(\infty,1)$-category of $H$-spaces.

For the sake of keeping things simple, I’m just going to talk about the $0$-space $E_0(X)$ in what follows; everything should work with $E_m(X)$ too.

Restricting $E_0$ to the orbit category $O_G$ gives a presheaf of spaces on $O_G$. Using Elmendorf, I’ll convert this into a $G$-space which I’ll call $F$. Thus $F^H$ is weakly equivalent to $E_0(G/H)$, and thus “points” of $F^H$ correspond to objects in “$H$-spaces”, or equivalently “$G$-spaces over $G/H$”. More generally, for any $G$-space $X$, the space $\mathrm{Hom}_G(X,F)$ is weakly equivalent to $E_0(X)$, so a “point” in this mapping space corresponds to an object in “$G$-spaces over $X$”.

What I want to claim is that $F$ is an algebra for the operad $U$. I won’t prove this; I’ll merely suggest how to think about the structure maps $(U(n)\times F^n)/\Sigma_n \to F$. Taking $H$-fixed points, these are maps

(1)$\psi_n^H: [(U(n)\times F^n)/\Sigma_n]^H \to F^H.$

Recall that the left hand side is a bundle over $[U(n)/\Sigma_n]^H$, that points in $[U(n)/\Sigma_n]^H$ correspond to functors $\rho \colon H\to \Sigma_n$, and that the fiber of the bundle over $\rho$ looks like $\mathrm{Hom}_G(S_\rho, F) \approx E_0(S_\rho)$.

Putting this all together, the map $\psi_n^H$ appears to take as input (i) a homomorphism $\rho \colon H\to \Sigma_n$, and (ii) an object $X$ in “$G$-spaces over $S_\rho$”, and gives as output something I’ll call $Y$, which should be a “$G$-space over $G/H$”. So I need to tell you how to get a $Y$ from $(\rho,X)$.

Remember that there is a map $f:S_\rho \to G/H$. Given $X\to S_\rho$, I want to let $Y$ be the “direct image” of $X$ along $f$. That is, $Y = f_!X$, where $f_!\colon (G\text{-spaces})/S_\rho \to (G\text{-spaces})/(G/H)$ is the right adjoint to the functor defined by pullback along $f$.

Examples:

If $\rho(H)=\{e\}$, then an object $X$ in “$G$-spaces over $S_\rho$” amounts to an $n$-tuple $X_1,\dots,X_n$ of $H$-spaces. The direct image $f_!X$ corresponds to the $H$-space which is the product $X_1\times \cdots \times X_n$ in $H$-spaces.

If $\rho$ is the left action of $H$ on itself, then an object $X$ in “$G$-spaces over $S_\rho$” amounts to a space $Z$ (with an action by the trivial group $\{e\}$). The direct image $f_!X$ corresponds to the $H$-space $N^H(Z) := \prod_H Z$, where the $H$-action is the one that permutes the factors.

Posted by: Charles Rezk on January 26, 2010 7:29 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I’m not quite ready to tackle ringed $(\infty,1)$-topoi yet, but I want to comment on this. I don’t think that adjunction can be true, because on the right you have something looking kind of like naive $G$-spectra (is the stabilization of a presheaf $(\infty,1)$-category the same as the category of presheaves of spectra?) and on the left you have genuine $G$-spectra. But a genuine $G$-spectrum is not just a particular kind of naive one; it has more data.

I think the way I would attack this question is to go back to the proof of the stable Giraud theorem and see where the domain $(\infty,1)$-category of the presheaf category that’s getting localized comes from. Then see what comes out when we apply that to the genuine equivariant stable category.

Posted by: Mike Shulman on January 25, 2010 3:07 AM | Permalink | PGP Sig | Reply to this

### Re: Equivariant Stable Homotopy Theory

(is the stabilization of a presheaf (∞,1)-category the same as the category of presheaves of spectra?)

Yes, that observation is part of the proof of the stable Giraud theorem, it is recalled as the first of the displayed equations in our section on stable Giraud.

I’d think this follows pretty directly from observing that loop space and suspension objects of presheaves are done objectwise, being pullbacks and pushouts. So also the stabilization works objectwise.

But a genuine $G$-spectrum is not just a particular kind of naive one; it has more data.

Hm, right. But can you help me see this more concretely: how would my functor from left to right (which I just sketched, I am not fully sure here) fail to be full and faithful?

Posted by: Urs Schreiber on January 25, 2010 3:54 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

how would my functor from left to right … fail to be full and faithful?

How would it succeed to be full and faithful? (-: I don’t see any reason to believe that it would be.

Posted by: Mike Shulman on January 25, 2010 10:40 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

How would it succeed to be full and faithful? (-: I don’t see any reason to believe that it would be.

But the stable Giraud theorem combined with Schwede-Schipley guarantees that there is some functor that embeds the $(\infty,1)$-categor presented by $Sp Cat(O^{st}_G, Sp)$ reflectively into that presented by $sSet Cat(O_G, Sp)$.

I don’t claim to see how this works in detail. But given that we know it must work somehow, I was just making an obvious guess what that functor could be. I currently don’t see how one would go about proving much at all about this “obvious” functor. But then, I haven’t really stared at it that long, either.

So if you can see concretely why that obvious guess can’t work, I’d be interested. Some other guess then must work!

Posted by: Urs Schreiber on January 26, 2010 12:08 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

But the stable Giraud theorem combined with Schwede-Schipley guarantees that there is some functor that embeds the $(\infty,1)$-category presented by $SpCat(O_G^{st},Sp)$ reflectively into that presented by $sSetCat(O_G,Sp)$.

The stable Giraud theorem as stated on the nLab doesn’t say what the domain $(\infty,1)$-category $E$ of the presheaf category involved is. Why does it have to be $O_G$ in this case?

Posted by: Mike Shulman on January 27, 2010 3:27 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I have a naive question: why only study actions of groups? What I mean is a group action is a special kind of a groupoid, an action groupoid. What about a more general groupoids, such as Lie groupoids or their topological analogues (I am not sure what that would mean concretely) or, even more generally, some sort of a geometric stack?

Stable homotopy theory of $G$-action groupoids, $G$ a fixed group, sounds very restrictive to me. I am sure there are historical reasons for it, but are there any mathematical reasons for being so focused on actions of groups at the exclusions of other groupoids?

Posted by: Eugene Lerman on January 25, 2010 6:43 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

why only study actions of groups? What I mean is a group action is a special kind of a groupoid, an action groupoid. What about a more general groupoids, such as Lie groupoids or their topological analogues (I am not sure what that would mean concretely) or, even more generally, some sort of a geometric stack?

The notion of “naive” $G$-equivartiant spectrum (the obvious definition that comes to mind) has a straightforward generalization to groupoids and beyond.

But historically people at some point decided that naive spectra are too naive, and that something more non-naive is needed. Unfortunately the non-naive version is a bit harder to interpret conceptually. It currently seems to justify its existence mainly due to the fact that it leads to interesting structures.

What we are discussing here is about how one might understand why “non-naive” genuine $G$-spectra are conceptually natural. Charles Rezk proposed that they can be understood as arising from a universal stabilization subject to a certain constraint on a monoidal structure. If that’s the right answer, I am still hoping that we can understand this as an analog of stabilization at geometric “Tate spheres”.

In any case, I gather it is not quite clear yet. When it becomes clear, when we have a good conceptual understanding on “genuine” $G$-spectra I suppose it will be just as straightforward to generalize to groupoid-equivariance as it is for the naive version.

So while I can’t help at the moment with groupoid equivariant spectra, I can point out something about this:

What about a more general groupoids, such as Lie groupoids or their topological analogues (I am not sure what that would mean concretely) or, even more generally, some sort of a geometric stack?

So here is a text that discusses the generalization of genuine $G$-equivariant spectra to the case that $G$ is generalized to an $\infty$-stack of $\infty$-groups modeled on $E_\infty$-rings (how’s that?):

I have maybe not fully absorbed this article yet, but I think I understand what’s going on. It looks to me – but I’d be glad to be corrected – that while this does enormously generalize the classical theory of genuine $G$-equivariant spectra, it does not really shed much light on the conceptual questions we were trying to understand. But maybe I am wrong about that.

By the way, concerning our other discussion about quasicoherent $\infty$-stacks of modules, there is some interesting stuff about this in the last two sections.

Posted by: Urs Schreiber on January 29, 2010 1:55 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Thanks.

Shortly after I posted my question here I did wander down the hall a few meters and asked the Burt Guillou more or less the same question. He wasn’t sure.

I suppose I could try walking down one flight of stairs and asking Charles Rezk, but that would be silly. After all, posting silly questions on this blog is a lot more fun.

Posted by: Eugene Lerman on January 30, 2010 1:41 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Over lunch today I kindly received helpful further input. When mentioning my suggestion that genuine $G$-equivariant spectra might be obtained from a localization of simplicial presheaves not just on the orbit category but on the category of all (sufficiently nice) $G$-spaces, I was being pointed to work by Andrew Blumberg, saying that pretty much this is achieved there.

I still have to look at this in more detail, but it seems that the article in question is

Andrew Blumberg, Continuous functors as a model for the equivariant stable homotopy category (arXiv:math.AT/0505512)

Quoting from the introduction:

we […] study […] equivariant diagram spectra indexed on the category $\mathcal{W}_G$ of based $G$–spaces homeomorphic to finite $G$–CW–complexes for a compact Lie group $G$. Using the machinery of Mandell–May–Schwede–Shipley, we show that there is a stable model structure on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal $G$–spectra. We construct a second “absolute” stable model structure which is Quillen equivalent to the stable model structure.

The full statement is Theorem 1.3, page 7

The “relative” model structure on the category of something close to simplicial (co)presheaves on the category of (nice) $G$-spaces is in def 2.3, page 10: the weak equivalences are those morphisms of (co)presheaves that become weak equivalences when evaluated on a geometric sphere, i.e. on an $S^V$ for $V$ a real linear $G$-representation.

This is then stabilized to a stable model category by taking the weak equivalences to be those that induce isos on $G$-spectrum himotopy groups on associated $G$-spectra.

I am not sure yet if this can be taken as the answer to what I had in mind. But it seems to be getting closer.

The article also mentions, by the way, a $G$-equivariant version of the May recognition principle, that we talked about with Peter May on the $n$Forum here. It is attributed there to

Costenoble and Warner, Fixed set systems of equivariant infinite loop spaces (JSTOR)

But I haven’t looked at this yet, maybe I am misunderstanding what is being attributed to whom here.

Posted by: Urs Schreiber on January 26, 2010 2:37 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Blumberg’s paper looks nice. I’d say what he does is something like this: he identifies equivariant spectra as being functors on (nice) G-spaces which make each G-orbit G/H “look like a dualizable object”. (That’s my interpretation of his Theorem 1.2, anyway.) It’s already known that “inverting representation spheres S^V” makes “G/H dualizable”, so Blumberg seems to have proved the reverse.

That’s really only a new observation for general compact Lie groups G (I think). For finite groups G, equivariant stable homotopy makes G/H look self-dual; this fact about equivariant stable homotopy for finite groups is basic thing that makes the theory of “G-equivariant Gamma-spaces” (as developed by May, Shimakawa, …) work.

Yes, the paper by Costenoble and Waner is the one which talks about “G-operads”, and proves a recognition principle for equivarant loop spaces (for finite G.)

Posted by: Charles Rezk on January 26, 2010 3:54 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

That’s really only a new observation for general compact Lie groups $G$ (I think). For finite groups $G$, equivariant stable homotopy makes $G/H$ look self-dual; this fact about equivariant stable homotopy for finite groups is basic thing that makes the theory of “$G$-equivariant Gamma-spaces” (as developed by May, Shimakawa, …) work.

i see. I have to run now and need to look into this later in more detail.

But can you rephrase this as something like: the stable $G$-equivariant homtopy category is that of the localization of the projective model $Func(G Space_{nice}, sSet)$ at the set of morphisms $xyz$?

Posted by: Urs Schreiber on January 26, 2010 5:59 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I think your rephrasing is correct, where $GSpace_{nice}$ means something like “$G$-spaces which are homeomorphic to finite $G$-CW-complexes” . Blumberg calls this category $W_G$, and for him its an enriched category (enriched over $G$-spaces); his functors are enriched functors, and the target category should be $G$-spaces. (I gather that this can always be rephrased without using $G$-enirched stuff; I think it’s equivalent to use the version of $W_G$ enriched over ordinary spaces (no $G$-action), and ordinary continuous functors to $G$-spaces. But don’t quote me on that …)

I don’t think this claim is very new, though quite possibly it hasn’t appeared in print before. I expect it was already known to experts that if you take functors from $\mathcal{W}_G$ to $G$-spaces, and you localize at a set of maps whose role is to “force representation spheres $S^V$ to be invertible”, then you get the stable homotopy category. (But I’m not an expert, so don’t quote me on that either.)

Mandell and May do something very like this in their paper on orthogonal equivariant spectra, except they use a different category for $GSpace_{nice}$ (a non-full subcategory of $W_G$ constructed from representation spheres).

Blumberg is working with $W_G$, and is proposing a different set of maps to localize with respect to, apparently ones which “make $G/H$ dualizable”.

(I think people have avoided dealing with $W_G$, because they believe that it doesn’t lead to a good model for commutive ring spectra. At least that’s what Mandell, May, Schwede, and Shipley seem to think happens in the non-equivariant case.)

Posted by: Charles Rezk on January 26, 2010 8:33 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I think your rephrasing is correct,

Thanks, but I am afraid I still don’t understand what I am trying to understand.

One aspect that doesn’t seem to fit what I am hoping to see is that Blumberg and his forerunners us copresheaves/diagrams also with values in $G$-spaces, instead of with values in plain spaces/simplicial sets.

But maybe I should say more explicitly what it is that I am trying to understand.

Namely, I am trying to understand the $G$-equivariant case as the generalization of the following non-equivariant situation (which possibly I also don’t fully understand…):

$\array{ Top \simeq Sh_{(\infty,1)}(*) &\stackrel{localize at \{\mathbb{R}\times X \to X\}}{\leftarrow} & Sh_{(\infty,1)}(Top) \\ {}^{\mathllap{invert categorical looping}}\downarrow && \downarrow^{\mathrlap{invert geometric looping}} \\ Sp &\stackrel{localize at \{\mathbb{R}\times X \to X\}}{\leftarrow}& ??? }$

Here on the top right we have $\infty$-stacks on $Top$. Localizing that at $\mathbb{R}$-homtopies yields plain $Top$ (as in Dugger’s notes). Inverting categorical looping in there yields spectra.

But as long as we are still in $Sh_{(\infty,1)}(Top)$ we can and should actually invert geometric looping, in terms of the geometric loop built from $\mathbb{R}$. Because that’s what we know is the right thing to do in other cases like motivic cohomology and, as we seem to have said, should also be the step that explains the “genuine” $G$-spectra.

My trouble starts probably with me not being quite sure how to think of the thing in the bottom right corner obtained this way. But I would expect that localizing it at $\mathbb{R}$-homotopies now turns this into the category of spectra.

Maybe I am wrong about this. With $I$ a line object in an $(\infty,1)$-topos, does localizing at $I$-homotopies commute with inverting $I$-looping?

In any case, my idea would be that we should try to understand the right column of this diagram for the equivariant case, because that seems to be where the geometric looping is natural, whereas after localization it looks like an ad-hoc construction.

So I am looking for a way to make sense o a diagram of the form

$\array{ G Space \simeq PSh_{(\infty,1)}(O_G^{op}) &\stackrel{localize at ??something??}{\leftarrow} & P Sh_{(\infty,1)}(G Space^{op}) \\ {}^{\mathllap{}}\downarrow && \downarrow^{\mathrlap{invert geometric looping}} \\ G Sp &\stackrel{localize at ??something??}{\leftarrow}& ??? }$

All diagrams here in big imaginary quotation marks. This is not a claim, but a question.

The thing is that in the $G$-equivariant literature we see the left side of these diagrams discussed. But from $\mathbb{A}^1$-homotopy reasoning etc. we know we want to be working on the right, and we also said that this would explain the non-categorical looping taken on the left, because that should really be the image of the $\mathbb{A}^1$-type geometric looping on the right.

So that’s the picture I am imagining, and of which i am trying to see if or how it is true.

Posted by: Urs Schreiber on January 27, 2010 2:06 AM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I wrote above (pretty imprecisely, but still):

$G Space \simeq PSh_{(\infty,1)}(O_G^{op}) \stackrel{localize at ??something??}{\leftarrow} PSh_{(\infty,1)}(G Space^{op})$

I see now that the answer (to the question marks) is on p. 50 of

• Morel, Voevodsky, $A^1$-homotopy theory of schemes (pdf).

Take the site $G Top$ to be the category of $G$-spaces that admit $G$-equivariant open covers (def 3.3.1) with the Grothendieck topology given by covering maps $Y \to X$ that admit $G$-equivariant splittings over such $G$-equivariant covers.

Let $Sh_{(\infty,1)}(G Top) := (sSh(G Top)_{loc})^{\circ}$ be the corresponding category of $\infty$-sheaves.

This has the obvious interval object $I$, the unit interval equipped with the trivial $G$-action (I suppose we could just as well take the full real line).

Then do the left Bousfield localization of $sSh(G Top)_{loc}$ at the collection of morphisms of the form $\{X \stackrel{Id \times 0}{\to} X \times I\}$. The homotopy category of the resulting model category is the standard $G$-equivariant homotopy category $G Top_{loc}$.

Posted by: Urs Schreiber on January 29, 2010 8:25 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

Given what I said above, I figured that if there is an answer along such lines, then it should have shown up in equivariant motivic cohomology.

I probably need to search around a bit more. This here is the first hit on a famous search engine:

• Ben Williams, Equivariant Motivic Cohomology (pdf)

If I see correctly, the definition of equivariant motivic cohomology on p. 5 is Borel equivariant motivic cohomology, namely the cohomology of the action groupoid.

Hm…

Posted by: Urs Schreiber on January 27, 2010 10:58 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

I wrote:

the definition of equivariant motivic cohomology on p. 5 is Borel equivariant motivic cohomology

The right definition of genuine motivic $G$-spectra is probably that in section 4.5, p. 8 of

• B. Guillou, A short note on models for equivariant homotopy theory (pdf)

I am glad I came back to this one. After I cited this at equivariant homotopy theory I had forgotten to read it to the end!

This looks like it is getting closer to providing the answer that I am after.

Posted by: Urs Schreiber on January 28, 2010 3:54 PM | Permalink | Reply to this

### Re: Equivariant Stable Homotopy Theory

In case anyone is interested, here are some notes from an introductory talk about equivariant stable homotopy theory, given by Anna Marie Bohmann at Chicago last week.

Posted by: Mike Shulman on January 26, 2010 9:06 PM | Permalink | Reply to this

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