@DenisNardin Are you in Regensburg now? Come say hi sometime...M305 upstairs

@AlexanderCampbell good point, I'm not sure

I may have asked about this some time ago already. Is there anything in the literature about "genuine operads" in the sense that the symmetric action on the space of $n$-ary operations $O(n)$ is genuine? So maybe define "genuine operads" as algebras for the composition product (which I assume exists) on "genuine symmetric sequences".

@SaalHardali What does it mean that the action is "genuine"?

@CharlesRezk I just mean $\Sigma_n$-equivariant space. So instead of a sequence of spaces with a homotopy action of $\Sigma_n$ you get a sequence indexed by $n$ where in $n$ you have a genuine $\Sigma_n$-space (i.e. presheaf on the orbit category)

Ah. Pretty much every treatment of operads in homotopy theory (even $G$-equivariant operads) assumes that $\Sigma_n$ acts freely. I actually tried to handle the more general case ($\Sigma_n$ not acting freely) in my thesis, but I would not now vouch for the correctness of anything I might have said about that.

In any case, the orbit category might not be the right thing to think about here. You really want to work with the following notion of weak equivalence $A\to B$ between symmetric sequences: they are equivalent iff for every cofibrant space $X$, $A_n\times_{\Sigma_n} X^n\to B_n\times_{\Sigma_n} X^n$ is a weak equivalence.

I think this turns out to be a weaker notion than equivariant weak equivalence.

I think the only relevant subgroups of $\Sigma_n$ turn out to be the ones that are stabilizers of diagonals in $X^n$. If I remember correctly.

I agree that in the end you'll probably only be interested in a further localization and not the full genuine structure

Somebody may have thought about it, but I'm don't remember seeing it.

I thought about this in the context where if you have an operad in spectra you can consider the symmetric sequence as a sequence of genuine equivariant spectra (i.e. borel equivariant spectra). Then you can take E-homology groups for some nice homology theory to obtain a sequence of mackey functors and from what it looks like you can actually get a sort of operad in $E_*E$-comodules whose components are mackey functors.

I was wondering whether one could develop a theory of algebras over these gadgets which could be a much closer approximation than algebras over the homology of the operad

The jist is I'd like to use information about the symmetric sequence of mackey functors you get out of an operad (and the natural induced maps between them coming from the operad structure) to understand better homology of algebras over said operad.

Sounds like you are asking if, on some symmetric monoidal model category of spectra, the functor $X\mapsto X^{\wedge n}_H$ lifts to the homotopy category, where $H\leq \Sigma_n$ is any subgroup.

hmm i'm unsure about whether this is what i'm asking... why do you think that? Even if it is it might be the case that the only relevant subgroups are stabilizers of diagonals (like you said).

What I do need is the genuine version of raising to a power which goes $(-)^{\times n} : Sp \to Sp^{\Sigma_n}$ which I think exist although I'm not entirely sure.

Consider the "genuine" symmetric sequence $A$ with $A_n=\Sigma^\infty_+(\Sigma_n/H)$, You want to turn this into a functor on spectra, which should really be a functor of $\infty$-categories. It should send $X$ to "$X^{\wedge n}_H$", whatever that means. I'm wondering what it means.

Ah, we might be thinking about the same thing.

The 0th step is finding an equivariant power functor for spaces $(-)^{\times n} : \mathcal{S} \to \mathcal{S}^{\Sigma_n}$.

@SaalHardali Here's the last time Saal brought this up for reference

thx ^^

For some reason I was under the impression that the equivariant power functor for spaces is an established thing but now I realize I don't really know why that's true and I don't have a reference

What would you want the "genuine power functor" on spaces to be?

er

maybe it's clear

There's a functor on 1-categories which takes a topological space and raises it to a power

It's clear this is equivariant in the strictest sense

The genuine power for spaces is easy to define, that's not an issue

Great. That's how i felt but my argument is not really nice

it sends X to the genuine Σ_n-space whose H-fixed points are X^{n/H}

You gotta work a bit more carefully to turn this into a precise definition, but it's not too hard

It's essentially a right Kan extension along a certain functor I→O_{Σ_n}^{op} of the constant functor at X

Yeah that's what I thought now

And I is just the category of points of presheaf over O_{Σ_n} corresponding to the Σ_n-set {1,...n}

I can believe there are several way of doing this. Is there a way to do an equivariant power for spectra?

That's known as the Hill-Hopkins-Ravenel norm :)

hmmm

Actually that makes sense

In this special case, it's probably easiest to define as the unique symmetric monoidal functor Sp→Sp^{Σ_n} sending Σ^∞_+X to Σ^∞_+X^n and commuting with sifted colimits

This is enough to specify it, because Sp=Space[(S^1)^{-1}]

I have a feeling I already went over this conversation with someone

I can't remember where it stopped though...

Thanks for the help btw

I have nothing intelligent to say about genuine operads unfortunately

IS there a composition monoidal structure on genuine symmetric sequences?

Not that I know of

So this is the first thing that's unclear here looks like

Is a genuine symmetric sequence just a seuqence $(X_n)_{n \in \mathbb N}$ where each $X_n$ is a genuine $\Sigma_n$-space?

Or are there compatibilities?

Yeah that's what I mean

No compatibilities

same definition for spectra

I might speculate that some additional compatibilities might be desireable. For instance, if you think of a symmetric sequence as a $B \Sigma$-space where $\Sigma$ is the groupoid of finite sets, you might want to "genuineify" the whole groupoid action...

But I don't know what that would mean

I'm thinking about situation where I apply a certain functor on an operad and that functor doesn't commute with taking homotopy quoteints. In this case considering the full genuine operad will remember more information even if you are only interested about the original non-genuine operad.

@SaalHardali The kind of "homotopy quotient-non-preserving" functor you have in mind is something like taking $E$-homology for some $E$, right?

Yes and variants of that

It could also be just a localization though.

Is there an analogous case you have in mind where "genuinifying leads to better approximations"?

The dream is to have an algebraic theory of algebras over a "mackey operad" (the thing you get by taking the homology of a genuine operad coming from an ordinary operad). Then these might help you understand better the homology of algebras over the ordinary operad.

Maybe even an obstruction theory

I think an example of what kind of information the "mackey operad" will be able to remember is Rezk's $\Gamma$-ring of power operations for example.

Because of Dold-Kan, simplicial abelian groups fail to model grouplike E-oo spaces.

What goes wrong trying to model E-oo spaces as simplicial commutative monoids?

@WilliamBalderrama Maybe people here will give more enlightened answers, but I guess my position is that strictification is the exception, not the rule. I would argue that the real surprise is that you can model E_1-spaces with simplicial monoids, not that you cannot model E_∞-spaces with simplicial commutative monoids

@DenisNardin I certainly agree with that. That's why I expect you cannot model E-oo spaces with simplicial commutative monoids, I just don't know a proof.

@WilliamBalderrama Well, all group-like simplicial commutative monoids are (homotopy equivalent to) a product of Eilenberg-MacLane spaces

This is very much not true for group-like E_∞-spaces (quick counterexamples: QS^0, BO×Z, etc.)

@WilliamBalderrama I asked this question once and got a nice answer in the comments from someone named Denis.... mathoverflow.net/questions/258549/…

oh -- maybe not precisely the same question

But the same idea holds -- a simplicial strict commutative monoid is a module over $\mathbb N$. Upon group completion, it becomes a module over $\mathbb Z$ and so is Eilenberg-MacLane as Denis says.

Well, yeah. I'm not claiming originality on that one :). It's a very old observation

@DenisNardin @TimCampion Aha, thanks. I was missing the reason why a grouplike simplicial commutative monoid would be equivalent to a product of EM spaces, I only knew this fact when the group part was also strict.

@WilliamBalderrama I think somewhere in Hatcher there's a fully elementary proof that a connected simplicial monoid is a product of EM spaces, and that's all you really need for the counterexample

Random question: I think I have the impression from somewhere that $I_{\mathbb Q / \mathbb Z} \wedge I_{\mathbb Q / \mathbb Z} = 0$. Did I get that right?

Yep, it's corollary 4K.7, only it's not super elementary and uses the Dold-Thom theorem

@TimCampion if that's the Brown-Comenetz spectrum, then this is in Bousfield's Boolean Algebra of Spectra paper.

I was thinking lemma 2.5, but it might not quite be a consequence of that, actually.

@WilliamBalderrama Thanks! It should be a consequence by taking $X$ to be the sphere, I think

@TimCampion Except the sphere doesn't have finite homology. I haven't looked at the proof to see where that's needed, though.

ah right

@DenisNardin with media exposure comes funding...

(along with general awareness)

@TimCampion if you work p-locally, then the homotopy groups of the BC-dual of the p-local sphere are all p-torsion, from which you can show that the BC-dual of the p-local sphere is Bousfield equivalent to the BC-dual of the mod p Moore spectrum. The latter satisfies $X \wedge X = 0$ by lemma 2.5 in Bousfield's paper, and therefore so does the former.

Then you have to convert to the integral setting...

@JohnPalmieri Thanks!