Which result are you talking about? I don't see how your counter-example applies.

no, what I said is not true as written.

that paper is fine.

Ah, ok. Sorry, I misunderstood.

I'm just wondering about some kind of general statement along these lines. His results seem to be only about stuff in homology, but is there some obvious action of SU(k) on CP^{n+k} for various n and k that gives these results on the nose (rather than in homology?)

i'm confusing myself: the functor $\mathbf{\Delta}^{op} \to \mathbf{\Lambda}^{op}$ isn't final, right? it's just $\mathbf{\Delta}^{op} \to \mathbf{\Delta}^{op}_\circlearrowleft$? (the last one is (the opposite of) the paracyclic category)

(and $\mathbf{\Lambda}$ is the cyclic category)

So at least you have an action of SU(k) acting on the first k coordinates of C^{n+k+1}, with fixed points C^{n+1}, right? Doesn't this give you what you want?

A naive question: There is an action of group G of power series with leading coefficient 1 on Lazard's ring L by universal property of L, and ANSS E_2 can be interpreted as group cohomology of G with coefficient in L as a G-module. Is there any way of making G acting on MU on the nose such that the homotopy fixed point is the sphere spectrum S so that homotopy fixed point spectral sequence and ANSS more or less coincide?

I have a bad feeling that it might be trivial somehow...but I still do not know yet...

@MingcongZeng I am a bit confused by what you mean. $G$ is an algebraic group and I do not really know what does it mean for an algebraic group to act on a spectrum. For what it is worth you can see the ANSS as the "homotopy fixed point spectral sequence" given by the coaction of $S[BU]$ on $MU$, but I'm not sure that's what you want.

Thinking about it twice I think I got something wrong for sure

The action I wish is $G$ maps to $Aut(MU)$ in a good category of spectra before passing to homotopy, as discrete groups. But even if it is true I doubt that we can define "homotopy fixed point spectrum" in this situation.

After all, $G$ acts on $\pi_*(MU)$ in a more or less evident way and I wish a lifting.

But $G$ is not a discrete group and the cohomology you are computing is not in the category of $G(Z)$-modules but of $G$-modules; that is comodules over $O(G)$

And the coaction by $O(G)$ is induced by the coaction of $S[BU]$, in the sense that $\pi_*(MU\otimes S[BU]) = MU_*\otimes O(G)$

(Here by $O(G)$ I mean the Hopf algebra over $Z$ associated to the algebraic group $G$, which as a ring is just $Z[b_1,b_2,\dots]$)

Oh right

I got it wrong

@EspenNielsen I guess just by passing to projective spaces?

So glad @DenisNardin is here fielding questions again. :)

Because I think that's roughly what I would have wanted to say, but I would take a hell of a lot longer to say it sensibly.

Hi @Jon. I actually thought if I should leave that one to you, I'm sure you have a better understanding than me of the relationship between $O(G)$ and $S[BU]$

Haha. Hm.... where $O(G)$ is the global sections?

I mean, right... I'm not sure.

Well over $Z$, it's just the Hopf algebra associated to the group scheme $G$

Right.

I think I'd like to claim that $S[BU]$ is somehow... something bigger.

I mean, it's clearly significantly more than just, say, a Eilenberg-MacLane type object.

And the point is, my current working theory at least, that it's something like the spectral bialgebra whose associated monoid is the automorphism group of complex orientations (aka topological formal group laws), or some such nonsense.

But I do think @MingcongZeng gets to something pretty deep. That is, what's the exact difference between a complex orientation of a ring spectrum $R$ and a formal group law on $R_\ast$?

Can someone please explain the kind of research done in dynamical systems theory?

@obe I fear you may not find anyone in here who even really knows?

But I'd be happy to proven wrong.

@Pieter what I would like to know is who starred your comment... (thanks for the link by the way, I had completely forgotten about it!)

terminology question: is there a name given to the kind of category where the composition of two morphisms is not a morphism but a set of morphisms?

@SaulGlasman is this a/k/a a Span(Set)-enriched category?

that's right

never seen em before

or heard a name

@SaulGlasman @RuneHaugseng do either of you happen to know the answer to my question above -- is $\mathbf{\Delta}^{op} \to \mathbf{\Lambda}^{op}$ final?

Sorry, I don't know much at all about cyclic things

haha yeah, me neither

@Aaron I do not think that it can be final, since it is not a weak equivalence

OHH duh

haha thanks

@SaulGlasman There's more than one way to make sense of that. For instance, we could equally well consider Rel-enriched categories, where Rel is the category of sets and relations.

right, the one I'm thinking of is enriched in spans of sets

although I'm doing it with oo-categories, so really it's enriched in spans X <- Y -> Z of spaces where Y -> X has discrete homotopy fibers

@AaronMazel-Gee the correct statement along these lines is that $\Delta^{op} \to \Delta_\circlearrowleft^{op}$ is cofinal, where $\Delta_\circlearrowleft^{op}$ is the paracyclic category

there's a proof in Rotational Invariance

I'm not aware of any name for such gadgets, but perhaps it might be interesting to point out that there's a notion of "category enriched over a bicategory" distinct from "category enriched over the underlying (2, 1)-category of a bicategory"

Proposition 4.2.8

@SaulGlasman @DenisNardin right, thanks. my confusion was because the colimit of a functor $\Lambda^{op} \to \mathrm{Spaces}$ naturally lives over $(\Lambda^{op})^{\mathrm{gpd}} \simeq BS^1$, so i was thinking that that was a (naive) $S^1$-space. in the end, my confusion was just that the un/straightening equivalence $\mathrm{LFib}(BS^1) \simeq \mathrm{Fun}(BS^1,\mathrm{Spaces})$ doesn't preserve the "underlying space"

right, when we talk about the realization of a cyclic space we mean the realization of the underlying simplicial space

which certainly is a naive S^1-space, but it's not the same as the colimit over Lambda^op

and actually it's really the S^1-action that captures the difference in colimits

@SaulGlasman can this notion of compositions being sets be related to the hyper-rings and hyper-groups of connes and consani?

i.e. sets where a+b is a set rather than another element.

possibly! reference?

in the sense that composition is a sort of "multiplication"

lemme look it up

i.e. a hyper-groupoid???? lol

@SaulGlasman what do you mean by this last line?

here's a starting point mathoverflow.net/questions/30288/…

here's a paper on it by a graduate student here at JHU

arxiv.org/pdf/1510.02979.pdf

here arxiv.org/pdf/1001.4260v2.pdf

@AaronMazel-Gee you can obtain $\Lambda$ by adding homotopies between morphisms to $\Delta_\circlearrowleft$ and it's those homotopies that end up giving you the circle action

if you took the colimit over $\Lambda^{op}$, you'd be going further and quotienting out that circle action

ah, okay. right, it's a localization

Paracyclic --> Lambda, i mean

here's a question: do you think it's possible to witness genuine-$S^1$ spaces as a reflective subcategory of Lambda-Spaces?

i think at least "proper" ones should be easy enough (i.e. remember all but the $S^1$-fixedpoints)

yes, you can do that pretty easily I think

how so?

at least if you're working in a point set world

because then you have the data of the C_n-fixed points of the nth level of your cyclic space

well, even if everything's $\infty$-categorical, you still have a $C_{n+1}$-action on $Z_n$ (for $Z : \Lambda^{op} \to \mathrm{Spaces}$)

you don't have the fixed points though. You can try to include them as data, and then you end up with the notion of "N-cyclic spaces", which figure in a paper Clark and I are currently writing

err... what are you asserting, exactly, anyways?

I hadn't finished

ok

haha sorry

no worries

if you let a weak equivalence of cyclic spaces be a map which induces weak equivalences of genuine C_n-spaces on the nth level, you end up with a model for the homotopy theory of genuine S^1-spaces for the family of finite subgroups

you can see this by looking at the way one gets the C_n-fixed points of the realization of a cyclic space:

you take the nth edgewise subdivision, so you end up with a cyclic object in C_n-spaces

and then you can take the fixed points levelwise and realize

right

I think this even works with cyclic sets rather than cyclic spaces

in which case you don't have to go out of your way to remember the fixed points

it's ultimately a special case, though

right, but I think you can model all S^1-finite subgroups-spaces with cyclic sets

yes

but so, what happens if you just take a "naive" cyclic space, i.e. a mere functor $\Lambda^{op} \to \mathrm{Spaces}$? you can still do the same construction, at least

yeah, but you no longer have any way of extracting the fixed points

"take the fixed points levelwise" ceases to make sense

well, you get a map $\mathrm{sd}_r(Z) : \Lambda^{op} \to \mathrm{Fun}(BC_r,\mathrm{Spaces})$. and we can still postcompose that with $\lim : \mathrm{Fun}(BC_r,\mathrm{Spaces}) \to \mathrm{Spaces}$

then you're choosing to use the homotopy fixed points

right

I don't think that'll allow you to represent all objects

okay

i mean, if this is going to be a left adjoint into genuine $S^1$-spaces, it'll have to be determined by its value on representables anyways. so that should be easy enough to check

okay, so this N-cyclic construction: can you pick up the $S^1$-fixedpoints too?

nope

how sad should this make me? it seems like plenty of things in the bokstedt/hesselholt/madsen/etc. world are only ultimately "$C_\infty$-equivalences" anyways

yeah, and I think that's all they're supposed to be

gotcha

so secretly the circle is always profinitely-completed anyways, and it's just a nice coincidence when something happens to be a true genuine $S^1$-equivalence?

I've seen the case made that K-theory should be thought of as the S^1-fixed points of THH, but I think that's more an aesthetic choice than a mathematical statement

yeah, I think for applications in things like THH, definitely

i thought it was TC that should be the S^1-fixedpoints of THH?

"should"

there are S^1-spectra over in the Survey of Elliptic Cohomology as well, and for those you really want to have the full S^1-fixed points

but they arise very differently

I think if I had to choose between K or TC being the S^1-fixed points of THH, I would choose K

what the heck is TC, really, anyways? how did BHM come up with that construction, of all things? i don't see how it'd come from an analogy with HC/HH

yeah, I'm guessing they just noticed that the trace could be lifted along Frobenius and Restriction

and used that observation to define TC

now we have the conceptual interpretation that TC is the mapping spectrum from the cyclotomic sphere into THH

see Blumberg-Mandell, "Homotopy theory of cyclotomic spectra"

right, but so then why cyclotomic spectra? i see that the "base case" of free loopspaces (+ nonabelian poincare duality) dictates that such structure should involve the geometric fixedpoints (as opposed to the categorical fixedpoints or whatever), but other than that i'm at a loss

good question

i guess i don't really know "why" THH is cyclotomic, is what i really mean

actually, I think the cyclotomic fixed points should factor through the S^1-fixed points, rather than the other way round

so K is definitely out as the S^1-fixed points

isn't there a slick factorization homology proof that THH is cyclotomic?

haha that's what i'm aiming for

but to do that, i need to understand "why cyclotomic"

you could try to interpret the geometric fixed points as a factorization homology pushforward along a self-covering map of S^1

or is that pushforward something else?

which pushforward are you referring to? in e.g. ginot's notes, he proves some sort of "projection formula", that for a factorization algebra $A$ on $X$ and a map $p : X \to Y$, $\int_X A \simeq \int_Y p_*A$ or something like that