@DenisNardin thanks again. So I am going to make a complete fool out of myself now: Can we express the map S^V_G(pt) --> Phi^H S^V_G(pt) explicitly as a matrix with respect to the direct summands of the tom Dieck splitting on both sides? What's the formula?

Actually let me think that through

No rush. Thanks a million for your help!

what is the algebraic K-theory spectrum of Z_p (p-completed if necessary) exactly? My memory tells me it's related to KU, but I don't know what's the statement exactly or where to find it

Bökstedt-Madsen 94 has the TC statement

yes I believe $K(\mathbb{Z}_p)\cong TC(\mathbb{Z})$ (mod $p$ coefficients for both)

yes, I just read that as well. cool. the precise statement is not that pretty but there's a bu there and an im J so I'm happy

Coming back to the equivariance:

if I may

Did some more exegesis of Lewis-May-Steinberger's section II.9 and have now written out the proof, completely in terms of their machinery, that the comparison map in question is just localization, up to equivalence. Now spelled out here:

ncatlab.org/nlab/show/…

In fact, its (cal-F[N]^prime)-localization, in their notation.

So I want to know when this map is surjective.

It's surjective when the localization functor is a full functor (on homotopy categories).

Hm, so when are localization functors full?!

I am thinking: At least whenever there is a minimum of calculus of fractions around, localization will be full: Because we keep everything the same except for declaring more maps to be equivalences, so the worst that can happen is that more cylinder objects and/or path space objects appear and make more morphisms equal.

Seems basic enough, but maybe I never seriously thought about fullness of localization functors.

I don't know. p-completion is a localization, which sends the sphere (with endomorphism ring Z) to the p-complete sphere (with endomorphism ring Z_p), so this localization is not full...?

@UrsSchreiber This argument would say that localization of rings are surjections. It clearly doesn't work even in the commutative case, and the problem is that when you force more things to be equivalences you are "adding" their inverses, which are new stuff

Hm, okay. Let's see, I also know that the localization is smashing...

that removes at least the counter-example of p-completion.

But not the p-localization one

Right, okay, thanks.

Well, at least smashing implies symmetric monoidal, so that I can now confidently fall back to the previous argument re traces, I suppose

What does Tanakian reconstruction give when feeded with the category of exact couples of vector spaces over a finite field?

@SaalHardali is this a Tannakian category? What's the fiber functor?

@JonathanBeardsley I was thinking of the forgetful functor. I'm not a hundred percent sure its tannakian (of course finiteness is needed for sure). I don't see any compelling reason for it not to be though.

I guess you have to have some kind of finiteness and spelling that out might rule out the examples that you find interesting.

And it isn't clear to me what you mean by forgetful functor. You have either two bigraded vector spaces with a map between them or one bigraded vector space with an endomorphism.

Yeah I'm not entirely clear on what the "underlying" vector space of an exact couple is.

@SeanTilson I realize this was very imprecise. When i said forgetful i was thinking about forgeting everything apart from the vector space (ungraded) which corresponds to the $E_1$-page.

I don't know that that would remember anything interesting...

Why won't it?

Maybe the better thing to do is really take the entire $E_1$ page with the bigrading like you said...

I mean, I guess you are asking "if I know the E_1 page as a vector space how much can I reconstruct from that?"

Oh, you were even going to forget the grading?

I guess... what is the kind of thing you would hope would be the answer?

My question is what can the endomorphisms of the fiber remember that sounds like a different question.

Or maybe i have this all wrong...

So an exact couple contains all of the information of the spectral sequence and I am guessing you would hope that just remembering the E_1 page might remember more structure than it ought to. But somehow the whole exact couple remembers implicitly non-abelian information (in the sense that an exact couple taking values in an abelian category can remember information ...

about homotopy theory over more complicated ring spectra than just HZ) but the E_1 page as just an F_q module has sort of no chance.

Maybe I am confused and you are realy interested in understanding what the relevant Hopf-Galois object is.

And I think the answer is something like "the information of the differentials" but I don't know how that organizes itself into an algebraic object.

Or at least the Hopf-Galois object contains that information.

That's preciselt the question. In slightly less formal terms i'm looking for the (pro? super?) algebraic group whose algebraic representations are spectral sequences.

Have you considered looking at the "Universal example" spectral sequences?

I don't know what you mean?

There is also a paper by Christensen-Frankland that talk about how all diffrentials relate to massey products in Adams type spectral sequences which might be relevant.

Sounds interesting I'll have a look, thanks.

So in whatever context you look at there are certain universal examples. They are spectral sequences that have exactly one nontrivial differential.

I think my original question still stands and is not total nosense.

Sorry, I didn't mean to imply anywhere that it was total nonsense but that I am not sure how useful the answer is/will be, does that make sense?

About what does tanakian reconstruction assign to the category of exact couples and the fuber functor which forgets everything but the $E_1$ page.

Maybe that is even a more hurtful thing to say. Regardless, being hurtful or rude was not my intention.

I understand your criticism, it wasn't too rude at all.

I guess you can build the grading into the group in the usual way.

Also, you might be interested in arxiv.org/abs/1805.00374

Well, it wasn't very friendly, and I apologize for that.

So I guess my reason for mentioning the universal examples is that the E_1 page just has two basis elements. And so distinguishing it as a spectral sequence from just the ss where the differentials are all 0 is the knowledge of that one differential. I guess that is obvious.

But then you see that since all these universal examples exist, and in fact generate spectral sequences in a certain imprecise sense (all differentials are detected by maps of exact couples from these universal example guys), then the knowledge of these differentials is the distinguishing data. I guess everything I just said is obvious.

But I don't know really what kinds of rules explain how different d_r's are supposed to interact. That doesn't... really come up much but if you want the Hopf-Galois object then you have to encode that I would imagine.

Maybe i'l explain where i'm coming from. The homology of an eilenberg maclane space with coefficients in a finite field is a cocommutative hopf algebra which could be interpreted as a sort of formal super group. This can lead one to a very accurate conjecture about the structure of the steenrod algebra (you probably familiar with this). I was trying to push the formal geometry analogies lately.

So I came to think about what happens to a postnikov tower of a space when i take homology. the homology of the associated graded pieces are the formal groups from earlier. Then they are glued in some complicated way.

Each filtration is glued to the next by a spectral sequence.

That's why i want an algebro geometric interpretation of this structure.

Basically the topological question i'm asking is what kind of algebraic(geometric?) structure do the homology of the associated graded pieces of a postnikov tower carry?

@SaalHardali very interesting. I don't know most formal geometry.

Have you looked at Hopf rings? Have you read Eric Peterson's course notes?

Does hopf ring refer to a specific paper/notes? Yeah eric peterson is awesome!

Maybe he's the guy to ask about this. I hope he sees it.

@EricPeterson Paging Dr. Peterson.

He talks about hopf rings in the course notes he is having published as a book on formal geometry and cobordism.

hello! i don't think i have anything helpful to say about this. i will say that i'm generally reluctant to admit that formal supergroups have any value, and also that stapleton, schlank, and i made a casual effort to give a formal-geometric description just of the E-theory of abelian 2-types (which is the babiest case of what you're describing), and we couldn't find a good answer

that definitely doesn't mean that there isn't one, and i think (modulo reference to supergroups) that it's a good question, just that i'm not the person to ask

good luck!

@EricPeterson Why don't you count the steenrod algebra as an example of where formal supergroups are helpful? (Just to make sure we mean the same thing - by "super something" i just mean the corresponding graded-commutative algebraic object corresponding to the "something" for example supergroup is a (formal dual) of a graded-commutative hopf algebra nothing fancier than this).

I mean the mod p dual steenrod algebra is very close to the automorphism group of a formal super group coming naturally from topology.

btw: i can't point to a precise connection, but this reminds me that i feel that there's a lot we don't understand about the relationship between the E-theory of finite complexes & the E-theory of pi-finite spaces, and there's probably a lot to mine here

there's a neat remark at the back of hovey-strickland that says that the p-adic K-theory of finite complexes always has a psi^p whose operator spectrum lives over some particular ring, and while the p-adic K-theory of BZ/p is a finite K^*-module, its psi^p has eigenvalues outside of this ring

I think this (about the relationship between pi finite and finite) is kind of a different way to state what i'm trying to get at.

What you mention sounds interesting

re: formal supergroups, my impression—which is wholly anecdotal and experience-based—is that formal supergeometry is only very exceptionally an accurate lens to view schematic results in homotopy theory. what is much more common (again, in my experience) is that some topological object has a natural filtration associated with it, the filtration pieces each have a natural formal-geometric interpretation, and occasionally the reconstruction data causes some homotopy lexseq to degenerate

the mod-p steenrod algebra is actually an example of this:

the stable BP cooperations, BP_* BP, are an even-concentrated ring that has a nice formal-geometric interpretation as automorphisms of a versal p-typical formal group; H_* BP arises as the quotient of BP_* BP by the sequence (eta_L p, eta_L v1, eta_L v2, ...) which is regular; and then H_* H /wants/ to be a quotient of that by (eta_R p, eta_R v1, eta_R v2, ...), but /that/ sequence is no longer regular because you already did the first quotient

so, even though that quotient is "what's happening in topology", what happens on homotopy groups is something different, because what used to look like quotienting by p turns into a funny steenrod extension instead

another example where you can see this happen is in the p-complete complex-orientable cohomology of BZ/p: for "most" choices of E, E^* BZ/p looks like E^* CP^infty / [p](x) for any choice of coordinate x; and for E = HFp, it just so happens that [p](x) = 0, and that causes the "super" class to appear

the super class isn't really there; it's an artifact of the p-series in Ga being weirdly small

this is also where the 'tau' cooperations come from in the Morava K-theory cooperations: the Morava E-theory cooperations are even, and then you have to double-quotient to get the K-theory ones and, oops, some information squirts around the edges of a lexseq into odd-dimensions again

people are entitled to different opinions on philosophical matters like this one, but this is mine

yeah, that's a faithful summary

Thanks for the comments!

(there's hubris in making any such claim about all of stable homotopy theory, but in the sub-realm of cooperations, i feel much more confident about making such a sweeping statement)

Yeah I intentionally made a very general statement to get a feel for what you're saying (sorry about that). It wasn't intended as an accurate summary ready for publication.

too late, i already sent you your ICM address invite

lol, I wish.