Here's a question. $GTop_*$ is based $G$-equivariant spaces, $C$ is presentable, stable, closed symmetric monodial. $L:GTop_*\to C$ is strongly monoidal and colimit preserving. Consider properties: (1) $L(S^V)$ is invertible for every representation $V$; (2) $L(G/H_+)$ is dualizable for every orbit. It's easy to show (1) implies (2); do we know that (2) implies (1)?
@CharlesRezk if you make G/H_+ dualizable, it seems like you get transfers, so your functor should factor through spectral Mackey functors, where rep spheres become invertible, right?
I agree that this is probably a version of "spectral Mackey functors"="equivariant spectra". Unfortunately, I have a hard time understanding why that theorem is true.
I think you can write down a proof along the lines I said, but it feels like overkill- it'd be nicer to just like check that that the evaluation map for the regular reps is an equivalence somehow
agreed
it seems like the way to go is by geometric fixed points
once you have monoidal geometric fixed points and you have the 'whitehead theorem' then you can reduce the claim to the trivial group
BTW, here's the proof that (1) implies (2), for finite $G$ anyway: To show $L(G/H_+)$ is dualizable, it's enough to give it the structure of a Frobenius algebra in the homotopy category of $C$. In fact, $G/H_+$ is almost a Frobenius algebra already: instead of the missing structure map, there is a collapse map $S^V\to G/H_+\wedge S^V$ for an embedding $G/H\subset V$. You can check explicitly that $G/H_+$ is a Frobenius algebra "up to $S^V$-suspension".
anyway, once finite G-sets are dualizable, and you're stable, then geometric fixed points are easy to define and the geometric whitehead theorem seems straightforward to check
I really just want to say that $G$-spectra are the universal thingy where finite $G$-complexes become dualizable. Then I can stop worrying about what $G$-equivariant stable homotopy theory is supposed to be.
Right, I think that's true... With the caveat that you probably want to remember that G/H_+ has half of a duality datum built in. You don't wanna mess that up when you make it dualizable
and I guess I think the way to formalize that is exactly this Mackey functor business
Oh but I was avoiding the stable Mackey thing. I'm saying: in the universal stable place where finite complexes become invertible, build a thing called geometric fixed points and check that they detect equivalences, by induction on the order of the group. Then check that rep spheres also became invertible in this place as I said above
Suppose C= the derived category of (algebraic) Mackey functors, so $L$ is the Bredon singular chains pushed into Mackey functors. What is the analogue of geometric fixed points there?
Probably just the derived category of abelian groups, but how do you get that if all you know is $C$?
All you need for geometric fixed points is a notion of categorical fixed points and to be tensored over G-spaces. In this case you have both- categorical fixed points are 'evaluate Mackey functors at G/H"
and the tensoring over G-spaces is via singular chains on a G-space
I'd rather say it this way: the composite $GTop_*\to Top_*\to Spectra$ which is "suspension spectrum of $H$-fixed points" is a monoidal functor to a stable category which takes finites to dualizables, so factors through our universal $L:GTop_*\to C$ to a functor $\Phi^H: C\to Spectra$.
and conservativity should now follow formally from the statement for G-spaces by writing everything as a hocolim of duals of finite complexes smashed with G-spaces
I don't quite see it, maybe because I don't really understand how isotropy separation works here.
We've got $\Phi^H: C\to Spectra$ which I defined above, and also fixed points $F^H:C\to Spectra$, defined by $F^H(X) := Map(L(G/H_+), X)$, where $Map$ is the function spectrum ($C$ is stable, so spectrally enriched).
Want to show $\Phi^H(X)=0$ implies $F^H(X)=0$, which presumably shows $X=0$, since the $G/H_+$ are generators for $GTop_*$, and using the universality of $L$.
We're supposed to have that $\Phi^e=F^e$, right? ($e$=trivial subgroup)
Classically, the geometric $G$-fixed points of $X$ are defined by a formula $F^G(L(\tilde{E}\mathcal{P})\wedge X)$, where $\tilde{E}\mathcal{P}$ is a certain explicit $G$-space. So I guess I need to know that that formula also computes the $\Phi^G$ I defined above.
i dunno... I guess I don't even know the classical proof of this. I guess it would actually be enough to check that it does the right thing on suspension spectra, but I don't see how to do that
But even classically, fixed points of suspension spectrum is complicated: it's the tom Dieck splitting! So basically that takes us right back to spectral Mackey functors ...
So $\tilde{E}\mathcal{P}$ is smash idempotent? I guess that's even true unstably. So there's a category of $\tilde{E}\mathcal{P}$-modules in $GTop_*$. Which I guess is the full subcategory of based $G$-spaces $X$ such that $X^H$ is contractible for all proper subgroups $H$.
shouldn't one actually want the condition (2') L(G/H_+) is self-dual? i think of spectral mackey functors as telling me that G-spectra is the universal stable place where the finite G-sets are self-dual, not just dualizable
i wish i understood better or more directly the G-spectra = spectra mackey functors equivalence, but my limited intuition at the moment is basically the calculation of the spaces appearing in the equivariant E_∞ operad (together with the equivariant recognition principle)
basically G-spectra must be the stabilization of G-E_∞ spaces, and G-E_∞ spaces are controlled by their associated lawvere theory, the full subcategory on the free objects, and this is equivalent to spans of finite g-sets
So there's an unstable category of $\tilde{E}\mathcal{P}$-modules in $GTop_*$. Which, if I'm not confused, is equivalent to plain old $Top_*$.
So if $L:GTop_*\to C$ is the universal dual thingy, we can likewise consider $L(\tilde{E}\mathcal{P})$-modules in $C$. That's a smashing localization, and we get a monoidal functor $C\to L(\tilde{E}\mathcal{P})Mod$.
On the other hand, I defined a $\phi^G$ above, induced from $GTop_*\to Spectra$ taking $X$ to suspension spectra of $G$-fixed points. This also kills $G/H_+$ when $H$ is proper. So we probably get a monoidal functor $L(\tilde{E}\mathcal{P})Mod\to Spectra$. Probably we want to show that is an equivalence.
Maybe we get lucky, and showing $L(\tilde{E}\mathcal{P})Mod\to Spectra$ is a formal consequence of universal properties.
@CharlesRezk I hope I did not miss anything in the previous conversation, but the theorem is not completely trivial. In fact I do not know of a nontrivial version of it for a compact Lie group
It is basically the collection of two remarks: finite G-sets are self-dual in orthogonal G-spectra (this is also known as the Wirthmüller isomorphism and it is a special case of Atiyah duality)
And that the tom Dieck splitting for finite G-sets holds both in spectral Mackey functors (basically by definition) and on orthogonal spectra (here you use an argument with the isotropy separation sequences like Dylan said)
My original question still stands: if we force finite G-sets to be dualizable (but perhaps not necessarily self-dual), do we end up with G-spectra? Does it even make sense to merely force objects to be dualizable?
My question is this: does there exist a presentable stable closed monoidal $C$, and a colimit preserving monoidal $L:GTop_*\to C$, such that the $L(G/H_+)$ are dualizable, but are not self dual?
Strongly monoidal functors preserve dualizable objects, because the data of "$A$ is dualizable" is in terms of that structure, i.e., $(B, A\otimes B\to 1, 1\to B\otimes A)$ satisfying some conditions. Furthermore, the data is essentially unique, up to unique isomorphism: dualizability really is a property.
So consider the category, whose objects are $(L:GTop_*\to C)$, where $C$ is presentable, stable, closed symmetric monoidal, and $L$ is a left adjoint and strongly monoidal, and such that $L(G/H_+)$ is dualizable in $C$. Maps $(L,C)\to (L',C')$ are $F\colon C\to C'$ are functors which are left adjoints, strongly monoidal, and with $FL=L'$.
Take the inverse limit in $\infty$-categories, giving $\hat{L}:GTop_*\to \hat{C}$. This will have $\hat{C}$ stable, cocomplete, symmetric monoidal with colimit preserving tensor product, and $\hat{L}$ strongly monoidal and colimit preserving, with the $\hat{L}(G/H_+)$ dualizable. I want to know if $\hat{C}$ is presentable, and even better that $\hat{C}$ is exactly $G$-spectra.
Aren't inverse limits of presentable categories automatically presentable? (although you run in some size issues but I think you can solve them by repeating the word "universe" often enough)
making sure i understand the situation: self-dualizability is also a property, right? and if i run the same construction with dualizability replaced by self-dualizability then i do get G-spectra for finite G?
I'm not sure that self-dualizability is a property. It should mean, for $A$ dualizable with dual $A^*$, that "$A$ is equivalent to $A^*$"; but you would want to witness this with a choice of equivalence $A\to A^*$, and that isn't unique.
@CharlesRezk But that is a problem, since that is not the correct map G/H_+ → (G/H_+)^∨ in G-spectra
@Arpon Yes, that is well known. A reference (I don't know if the first) is this paper by Blumberg
Uh assuming you want the target to be stable of course, otherwise there are easy counterexamples (connective G-spectra...)
Well, I guess I am still abusing the terminology.. When I say "self-dual" what I meant above is "self-dual with a very precise map realizing the duality"
@CharlesRezk ah ok, i suppose i can see why asking just for the existence of an equivalence of A and A^* would be bad for forming something like a limit. by the way, is your question about that limit basically asking if that category has an initial object (and if so what it is)?
I was saying (assuming I didn't mess up) that you can always take a limit in (possibly LARGE) infty-categories, and that this limit object inherits some of the desired properties. If it has all the properties (in particular, presentable), then its really an initial object of my diagram.
You want to take that limit in the (huge) ∞-category of symmetric monoidal stable presentable ∞-categories. There is an issue with the diagram itself being huge, which I am not sure how to fix without losing the automatic presentability, but I also feel this is not the interesting part of the question
