@RuneHaugseng yeah, that seems particularly trivial since isn't a commutative algebra in (A,B)-bimodules for A and B commutative just a commutative (A \otimes B)-algebra? so this would be some formal construction on the cocartesian symmetric monoidal category (CAlg,\otimes). i think this is ultimately just Spans in the opposite category.

i guess somehow what's more interesting about morita n-categories for n<∞ is that tensor product doesn't have an "easy" universal property, but rather has to do with enforcing commutativity that wasn't already automatic

@JonathanBeardsley i'm not sure what's going on here, but many of your symbols are just empty little boxes -- at least for me

@JonathanBeardsley this is interesting. it seems like there are actually perhaps two monoidal structures e.g. on the category of modules over a bialgebra in a braided category, depending on which you braid over the other? it feels like there should be an equivalence between the two possibilities, but i'm not seeing it.

is there a "moduli" description of modules over the cohomology ring of a space, perhaps under some assumptions? (e.g. simply-connected, or maybe even a manifold, or maybe i need to mean rational cohomology. i'm also happy to mean "cohomology ring spectrum" here.) i'm vaguely recalling something analogous to the fact that modules over $C_*(\Omega X)$ are the same thing as local systems over $X$.

@AaronMazel-Gee Good point! This indeed looks like it would be obtained from the Morita n-categories of commutative algebras in V (e.g. since the Boardman-Vogt tensor is commutative), which just gives n-fold cospans in CAlg(V). (We proved that in arxiv.org/abs/1904.11312) So the good news is such an $(\infty,\infty)$-category exists, but the bad news it's not a new example...

@AaronMazel-Gee sometimes it happens that Mod_{C^*(X)} = Fun(X,Mod_Z). For example, I think this is true for every simply connected finite space. This is equivalent to the global sections functor lim_X to be colimit preserving, that to an Eilenberg-Moore property for the inclusion of pairs of points into X. I think finiteness gives the colimit preservation of the global sections and the simply connectedness should give you the Eilenberg Moore property, but im not 100% sure.

If you are happy with examples that are not Z-linear, then every n-truncated p-finite space has this property in Sp_{K(n)} (Hopkins Lurie Ambi paper chapter 5 somewhere).

*and to an Eilenberg Moore property

@S.carmeli I think you need slightly more than just simply-connectedness for Eilenberg-Moore, you also need H^*(Omega X) to be (degreewise) of finite type

Maybe this follows from finiteness of X ?

ah yes it should, by the Serre spectral sequence I guess

@MaximeRamzi yes it is needed and it follows from the finiteness of $X$, thanks!

@S.carmeli Could you explain though how it follows from lim_X + Eilenberg-Moore ? From the finiteness of X I can deduce that the left adjoint to lim_X is fully faithful Mod_{C^*(X)} -> Fun(X,Mod_Z) but I'm not sure how to use Eilenberg-Moore to get that it's essentially surjective. Using EM I can see that C^*(Omega X) is in the image (it's the image of Z via a point inclusion Z ->C^*(X)), and I guess it's almost what I want because I only need Z[Omega X] to be in the image

@MaximeRamzi I hope im right about it, but i just mimic here an argument I learnt from Hopkins-Lurie paper:

first, note that we have a right adjoint functor p_*:Fun(X,C)<--->Mod_{C^*(X,1)}(C) for every presentable stable symmetric monoidal C where p:X-->pt the projection.

we assumed it is colimit preserving, and from this you can easily deduce it is also C linear. it has a C linear left adjoint given by the "localization functor" M|--> p^*M(x)_{p^*p_*1} 1

now you have to check that the unit and count are iso's and since all the functors are C-linear colimit preserving, its suffices to check on generators. For Fun(X,C) these are the skyscraper sheaves at points of X, and for Mod_{p_*1}(C) it is the free module p_*1. for the count at p_*1 the statement is clear and for the unit using base-change and projection formula it reduces to EM.

is that helpful?

wait, maybe we needed in their argument the fact that the loop space is also nice, sorry im might be completely wrong here!@

anyway, that was the idea, which no I'm not sure at all about

yes, I agree, i forgot that we needed in HL the weak-ambidexterity of the space, my fault :-(

@S.carmeli Ok so I had most of what you sat until the statement about the skyscraper sheaves. These look like Z[Map_X(x,-)] but I'm not sure what their limit is so I'm having a hard time seeing what the counit looks like for those

say*

@MaximeRamzi yes, me neither. too used to pi-finite spaces in Sp_{T(n)} :-)

Ha no problem ;) it'll give me something to think about

because well with no 1-connectedness assumption we do have a fully faithful embedding Mod_{C^*(X)} -> Fun(X,Mod_Z). With simply-connectedness it's not hard to show that the image contains Fun(X,Perf(Z)), but if Omega X is not finite (which I guess it rarely is when X is; I would imagine never but I'm not sure) then that doesn't contain all the compacts in Fun(X,Mod_Z)

@MaximeRamzi right. This point always confuses me. So it seems like modules over C^*(X,Z) wants to be some kind of renormalization of Fun(X,\Mod_Z) where we declare the local systems built from the trivial local system as compact generators or something?

yes, something like that

In fact, Mod_{C^*(X)} seems to be exactly Ind(Fun(X,Perf(Z)) when X is simply-connected

When X is not simply-connected, you replace Fun(X,Perf(Z)) with exactly what you said

Let G be a finite group such that every finitely generated ZG-module is nilpotent; does it follow that G is trivial ?

(if not, then if pi_1(X,x) = G we can still put Fun(X,Perf(Z)) which is a simpler description :) )

Ah yes : let p be a prime not dividing |G|, then F_p[G] has many irreducible representations

finite group? didn't we talk about finite spaces? I'm confused...

i mean shouldn't we consider finitely presented groups or something?

@S.carmeli Yes sorry I was trying to see if there was a more general class of groups for which Fun(X,Perf(Z)) consisted of exactly those local systems built from Z^triv

For instance if we replace Z with a field k of char p, then any X with pi_1 a p-group will have that property

But wait if I'm not mistaken we have the following thing : if X is finite and 1-connected, and Omega X is not finite in Z-homology (again, I'm not sure that doesn't follow from the other hypotheses), then the image of Mod_{C^*(Omega X)} is exactly Ind(Fun(X,Perf(Z))), but Fun(X,Perf(Z)) are not the compacts in Fun(X,Mod_Z) by assumption on Omega X, so this is never an equivalence

So I guess to answer Aaron, either he'll be happy with Ind(Fun(X,Perf(Z)), or we have to find some other approach

yes yes this shows it is basically never true, while I claimed it is always in the beginning :-|

Yup, well, that happens haha :D (to me, more often than I'd like to admit)

@AaronMazel-Gee Sorry it must be because I'm using unicode symbols for things like \mathbb{E}

@AaronMazel-Gee I don't think I entirely follow, but I think I can imagine an infinite number of monoidal structures based on that middle map $M\otimes N\otimes H\otimes H\simeq M\otimes H\otimes N\otimes H$, where I've applied the twist to the middle two terms.

Basically you could iterate that twist as many times as you like in a braided monoidal category and get a different isomorphism

Or, rather, not the twist itself, but the double twist.

$\tau_{NH}\circ\tau_{HN}\circ\tau_{NH}$

Of course if your category is symmetric monoidal then these just keep collapsing and you only get (up to equivalence) one monoidal structure on the category of modules over a bialgebra.

However, the way that one constructs the first monoidal structure that I described (where you just apply the twist once) is sort of canonical. It's not immediately clear to me how you would encode the iterated twist monoidal structures "fibrationally."

Maybe one can precompose with automorphisms of the $\mathbb{E}_2$-operad or something. I'm not sure.

Is localizing at Morava K-theory "compatible with the tensor product" on spectra in the sense of HA.2.2.1.9?

Or is being compatible in that way basically the same as being smashing?

@JonathanBeardsley every Bousfield localization is compatible with the symmetric monoidal structure, because the E-acyclic objects form a (x)-ideal so maps with E-acyclic fiber are closed under tensoring with arbitrary spectrum. But I hope I use the same definition of "compatible with the symmetric monoidal structure" as you refer to (I think I do but I lost faith with my understanding earlier today :-))

Well ultimately I want to know that if I have an E_k-cell attachment diagram and I Bousfield localize it, do I get another E_k-cell attachment diagram.

if you consider the result in the category Sp_E then yes, of course if you re-embedd in Sp then you are in trouble.

Right.

Hm... yeah I mean I know that localizations preserve colimits, but then I think I'm basically asking whether or not localizations lift to E_k-algebras and then still preserve colimits (of algebras)

they are colimit preserving and symmetric monoidal, so I guess its enough right?

Yeah, seems like it?

Well, maybe not I guess, since colimits of algebras aren't going to be computed in the underlying category?

I guess you could ask whether the lift to algebras is still a left adjoint.

no but e.g. for E_infty filtered colimits are fine and pushouts are also because they are given by Bar construction so they are also fine

oh sure also this way, a symmetric monoidal adjuntion gives an adjunction on E_k algebras, the right adjoint lifts by the lax structure on it

Aha yeah.... I'm very very anxious about generalizing statements from E_\infty to E_n

@S.carmeli ah yeah sure, that's nice.

Sorry right this is basically the higher algebra citation i gave above

I think David White has a concrete theorem about this in terms of monoidal model categories.

It seem to me like a conceptual argument for this should be something like a left adjoint symmetric monoidal functor C-->D should be a left adjoint functor C^(x)-->D^(x) in Op^\infty (since maps of operads C^(x)-->D^(x) is a lax functor) and hom(E_k,-) is an (infty,2)-functor from Op_infty to Cat_infty so it should preserve internal adjunctions? something like that.

well that's not a proof of course but an intuition

@RuneHaugseng any idea whether or not there is a notion of "cleavage" of a cartesian fibration of simplicial sets?

@S.carmeli @MaximeRamzi thank you both for all the comments, but i'm missing something very basic: what is the functor $Mod_{C^*(X)} \to Fun (X , Mod_Z)$? also, by the way, i'd even be happy with just the rational case.

@JonathanBeardsley haha yes right, by two i meant infinity. of course the whole theory of braided categories has a symmetry, so i'd say there is in some sense nothing more canonical about taking $\tau$ then $\tau^{-1}$.

@AaronMazel-Gee ah yeah I didn't even think about $\tau^{-1}$.

@AaronMazel-Gee its the left adjoint to the global sections functor. I.e. its something like M--> p^*(M)(x)_{p^*p_*Z} Z where p:X-->pt and Z is the constant local system on the integers.

and I don't think using Q instead helps much: it is still an issue that Fun(X,Mod_Q) is not generated under colimits from the constant local system Q I think.

but you know, double-check for your own safety:-)

@S.carmeli I think it might be generated by something like the objects Hℚ⊗Ω(Xₖ) where the Xₖ run through the connected components of X?

Sorry maybe this was already contained in the stuff that Maxime said...

I just remember I got confused about this recently.

well that was the heart of our previous discussion, the difference between that and HQ^{OmegaX_k}. I am too ambidextrous to see the difference already :-D

Oh sorry!

Yeah after writing it I was thinking "That comment was probably superfluous..."

no it wasnt! knowing im not the only one confused by it helps a lot thanks!

I should perhaps point out that there's sort of a tension between the finiteness of X and that of Omega X which makes it so that when X is finite, we have the situation we described earlier, but when Omega X is finite we have a sort of "dual" situation (in an absolutely not technical sense) where Fun(X, Perf(Z)) is the same as Perf(C^*(X))

(because we can still check that the unit is an equivalence on C^*(X) ! and so it will be one on the whole thick subcategory it generates, we use the finiteness of X to go from there to the localizing subcategory). In fact if we allow ourselves to change the base, we can relax the simply-connectedness assumption in the latter case.

and if Omega X is finite, then Perf(Z[Omega X] ) is included, in general strictly, in Fun(X,Perf(Z)), so Fun(X,Mod_Z) becomes included in Mod(C^*(X)), as opposed to the situation where X is finite

@MaximeRamzi yah, but not via p_* in this case but via the ind-completion of its restriction to the compact objects. I guess it should be something like "cohomology with compact support" in some vague sense.

Are there conditions I can put on a diagram of groups so that B of the colimit of the diagram is the hocolim of B of the groups? This is true, for example, for a quotient map, a free product, or more generally a pushout diagram where both (at least one?) of the maps is an injection.

I'd like to be able to talk about "free resolutions" of groups, and say that there's a corresponding resolution of group cohomology. It seems like a really simple idea but I can't find anything about it.

So, pi_1 of the hocolim is definitely the colimit of groups, so you're equivalently asking when we know that a colimit of 1-types is again 1-truncated. I think partial results are given by the chapter "Graphs of groups" in Hatcher (he doesn't phrase it in terms of colimits, but in terms of an explicit mapping cone construction) .

(roughly: If your diagram has the shape of a graph, and all morphisms are injective group homomorphisms, then it works. This generalizes the situation of a pushout of two injective maps)

B of the homotopy colimit in grouplike E_1-spaces will always be the hocolim of the diagram of B's in pointed spaces - because grouplike E_1-spaces are modeled by simplicial groups with underlying fibrations and weak equivalences, you're essentially asking a model categorical question : when is my colim a homotopy colim ?

So I guess you'd have to understand cofibrations between discrete groups in there