I think my plan was always to use a repository for my hypothetical thesis, and so one day out of procrastination probably I signed up and tried to learn how to use it
Is there any known theory of constructing moduli spaces for, say, all possible actions of a given monoid $G$ on an object $X$? That is, all possible group actions $G\times X\to X$ in some category?
it is a bit easier than that though, right? a monoid or group is a category with one object, a diagram is a functor from a category of a particular shape...
Does anyone know anything about Bousfields notes on simplicial commutative rings from Brandeis?
@SeanTilson I don't quite follow. I have a monoid $M$ in a simplicially enriched category $C$ and an object $X$. I can certainly see how I can get the moduli space by looking at a certain subspace of $C(M\times X,X)$, and by looking at it via some equalizers. But how is this related to $M$ being an internal $C$-category with 1-object
Like, how can I realize this as a map of diagrams in $C$?
Hm, I said equalizers, but perhaps I mean coequalizers.
That is, I've got two maps $C(M\times M\times X,X)\to C(M\times X,X)$, one given by $M$ acting on $X$, and one coming from the multiplication on $M$, and I want to coequalize these, etc.
Hm, yeah, I think perhaps these are the sorts of thing I want to think about, I just want to, in my particular case, replace their monad $T$ with a comonad. Interesting.
Wow, I don't know when this stuff of Johnson and Noel was done, but it either anticipates or works very nicely with Kathryn Hess' stuff.
I mean... they're totally computing descent spectral sequences... sort of. Or like, well, I should say descent spectral sequences are an example of what they're talking about.
Not. I'm trying to unwind this statement of AKhil: "(More generally: Given any symmetric monoidal category, if the unit object satisfies some categorical property, then so does any invertible object. This is useful in other contexts. Example: any invertible object in the stable homotopy category has to be a finite spectrum, because finite spectra are the compact objects; from here it's not too hard to conclude that the invertible objects in spectra are the spheres.)"
It'd be nice to figure out the exact argument because the usual proof the Pic(S) = Z is kind of hard.
idk if category theorists have coopted the word to mean something else, but i think compactness should mean that a map to a cofiltered colimit factors through the colimit of a finite stage
Yeah, all these things are given precise definitions in Axiomatic Stable Homotopy Theory (awesome book!!), but you never really know what people mean when you talk to them.
so for a map of rings R->S, we get a comonad by forgetting S-module structure then tensoring up with S again right?
there's a map M (x)_R S --> M (x)_S S since the tensor over S identifies strictly more stuff
i guess the slightly nicer thing to say is that there's a natural transformation of cospans [M (x) S <-left-act-- M (x) R (x) S --right-act-> M (x) S] ==> [M (x) S <-left-act-- M (x) S (x) S --right-act-> M (x) S] induced by applying f to the middle argument, and so a map of pushouts, which are the tensors over R and over S respectively
i think i am finally coming to terms with the operadic two-sided bar complex
idk, it's not, it's just highly misleading if you skip all the introductions to all the papers and keep saying 'ya sure i know what bar complex is, i know what tensors are, come on'
it is the standard resolution associated to the monad X |-> X ^ MU, which is the other order of applying those push-pull adjoints you were asking about a bit ago
what's an instructive example of a right module for an operad
if you have an algebra for an operad, you can build a left-module by taking the constant symmetric sequence at the algebra, and then the left-module structure maps look like P(j) (x) C^{(x) j}, which has a natural map to C using the algebra map
and i think if C is an operadic algebra, then you can make the sequence M(j) = Maps(C^{(x) j}; C) into both a left- and a right-module for the operad
but that's not very useful i feel for knowing how i should think of a right module
it would be excellent if a module M for an algebra A over an operad P could be made into a right P-module to sort of fill up the other slot, but i suspect that's not how this goes
progress! i've disconvinced myself that i understand the koszul morita equivalence functor in terms of operadic co/bar duality. i think out of chat.stackexchange.com/transcript/message/12220918#12220918 , 1 and 2 are captureable: the bar complex in 1 is the operadic bar complex B(1; P; 1), and the bar complex in 2 is B(1; P; A), where the P-algebra A is considered as a left-P-module. where does an A-module M fit into this? the classical bar complex B(k; A; M) seems a level removed...
Oh I think I said a lie. The notation K_0 was meant to denote the Grothendieck group ring of isomorphism classes of dualizable objects. Confusing notation.
Hi @Drew, I think the statement that the Picard group of the stable homotopy category is $\mathbb{Z}$ can be proved as follows: first, you argue that an invertible spectrum is compact, e.g., because the sphere is compact and tensoring with an invertible spectrum (as an autoequivalence) preserves compact objects. Now you need to show that the homology of your spectrum is $\mathbb{Z}$ concentrated in one dimension. So "tensor up" over $S^0 \to H \mathbb{Z}/p$ (i.e., consider mod $p$ homology).
Invertible objects in $H \mathbb{Z}/p$-modules are concentrated in one dimension since you have a K\"unneth theorem; now all that's left is to see that everything fits together at each prime. More generally, this argument can be used to show that the Picard group of modules over a connective $E_\infty$-ring (with $\pi_0 $ noetherian and connected, say) is given by the Picard group of $\pi_0$ times $\mathbb{Z}$ (for suspension).
One way to streamline this argument is to use the theory of flatness for modules over connective structured ring spectra. There's a nice criterion for flatness: if $A$ is an $A_\infty$-ring with $\pi_0 A$ noetherian commutative, then an $A$-module $M$ is flat if for every field $k$ equipped with a map $\pi_0 A \to k$, the relative tensor product $M \otimes_A k$ is discrete.
This lets you argue that, in the connective case, invertible objects are automatically flat (which means that the Picard group is "algebraic"). I think this argument (more or less) is in a paper by Baker-Richter "Invertible modules for commutative S-algebras with residue fields."
Hi @TylerLawson; I think you can argue this way: an element in $Pic( \pi_0 R)$ is projective, and any projective $\pi_* R$-module can be realized by an $R$-module (by explicitly writing it as a retract of a free $R$-module).
So in particular, I don't think one needs connectivity for this to happen.
Hm. That's a good question. I guess I'm thinking about this setting of Johnson and Noel's. Where they construct these obstruction spectral sequences for lifting maps up to maps of algebras over a given monad.
In particular, I think one can take the free/forgetful adjunction for having a $G$-action.
But, as I'm typing this, I'm not sure, I think I'm off somehow.
This spectral sequence they get seems like it should be the descent spectral sequence for some kind of generalized monadic descent situation.
Or rather, perhaps codescent. Since they're going from C(A,B) to C_T(TA,TB)
Whereas descent spectral sequences go the other direction.
Okay. I think I'm probably just on a role with saying things that make zero sense tonight. I mean, I frequently babble, but this is getting out of hand.
I need some sleep. And then coffee. And then I need to think hard for a day or two before saying anything else.
@JonBeardsley, I think there is an interpretation: if G is a finite group, the category of G-objects in a category is equivalent to the category of sheaves on the category of finite G-sets (where the topology is based on epimorphisms). The global sections are given by fixed points. If you do this in homotopy theory, I believe that the descent spectral sequence you get is the HFPSS.
@AkhilMathew Where if we call the category C, you mean C-valued sheaves?
Right, so we can think of global sections in this case as being something like descending along a cover of the terminal object with trivial G-action or something (I guess the point, in this case). But to do more intermediate pieces, we actually get an intermediate descent spectral sequence, something like a Galois correpondence.
Ah yes, where we look at fixed points under some subgroup. Fantastic.
Now, suppose we have a Galois extension of rings $R\to S$ by $G$. I want to say something like the category of descent data here is equivalent to the category of $S$-modules with $G$-action, but this is entirely by analogy, so I'm not sure that's accurate. The HFPSS for a module $M$ (with some given $G$-action, i.e. a descent datum) gives one the $R$-module that $M$ descends to. But we can think, as you say, of descent data as being something like (for a suitable site based on $G$)
$S$-mod valued sheaves.
And I want to say something like $R$ is the terminal object on this site. Hence what I'm really doing by computing HFPSS if evaluating my sheaf's global sections.
@JonBeardsley: the category of descent data is not quite that, since the G-action won't be in the category of S-modules; it'll, rather, be the homotopy fixed point category for the G-action on the category of S-modules.
E.g., in algebra, there's a complex conjugation on $\mathbb{C}$: this induces a $\mathbb{Z}/2$-action on the category of $\mathbb{C}$-vector spaces given by $V \mapsto \overline{V}$.
The homotopy fixed points for that action are given by the category of $\mathbb{R}$-vector spaces (by Galois descent).
Okay. Yeah, alright. Since Galois descent is faithfully flat, the category of descent data is going to be equivalent to the category of modules over the source of the extension.
I'm not sure where this is written down (Lennart and I make a quick reference to it), but it certainly the idea of Galois descent is very old (at least going back to Grothendieck), though algebraists probably don't say the phrase "homotopy fixed point category."
Here's how I'd think of the homotopy fixed points: if you have a set S, with an action of a group G, then the fixed points S^G are those s \in S with gs = s for each g \in G.
In other words, my category of descent data should be something vaguely reminiscent of either "matching families" or "coalgebras over a comonad" or something, so I'm just not seeing where that's coming from.
Now if you're thinking higher categorically, saying gs = s (where s is an object of some category) is not what you're supposed to do: instead you have to say that gs is isomorphic to s, and choose an isomorphism.
You also want to choose the isomorphisms gs \simeq s to satisfy some coherence (cocycle) conditions. The collection of all possible choices (objects s, plus isomorphisms as desired, plus coherence -- and even higher homotopies if you're in a $\infty$-category) is the homotopy fixed point category.
If $R \to S$ is $G$-Galois, then you can identify the descent data along that adjunction with the homotopy fixed points for $G$ acting on $S$-modules.
To see that, you have to identify the cosimplicial (cobar) complex you get and identify that with the cosimplicial construction for computing homotopy fixed points (using the Galois property).
In algebra, it'd mean the following: if G acts on a ring R, and M is an R-module, then for each g \in G, there is a new R-module M^g which has the same underlying abelian group, but such that the multiplication law is now r m = g(r) .m where the second is the old multiplication.
there's a category (Ring-Module pairs) with a forgetful functor down to rings
descent data is a lift of the functor to a functor BG -> (Ring-Module pairs), or equivalently a section of the pullback category (which is a category over BG whose fiber is the category of S-modules)
[and I believe that something like this is in the DAG chapter on quasicoherent sheaves]
@JonBeardsley -- to continue along the tack that @SeanTilson was taking: a monoid action of M on X is equivalently a monoid morphism M --> End(X), so you're asking about the space of map between two algebras (in spaces) over the associative operad. if you're willing to be \infty about it, you can use my GHOsT business for this...
i was hoping to dispense with it, but it ends up being totally crucial. i just ask for an "adams's condition" on the object E that's playing the role of "homology"
i think the weakest i can get away with is saying that my category has a monoidal structure which commutes with colimits in the first variable, defining E_*X = \pi_*(E \otimes X), and then saying that E satisfies a "left" Adams's condition
the statements about twisted forms are relative (for R -> S a finite G-Galois extension you can classify R-(xxx)'s that become equivalent to a fixed object after extending to S)
@AaronMazel-Gee I'm not 100% sure I follow, but you can use some kind of Pic-graded model structure if you need things where homotopy groups won't get it right sometimes
This seems like it might be a way of obtaining new elements of picard groups, if one were able to do this for more general... I dunno, profinite groups?
@TylerLawson the cellular motivic spectra are the ones that are generated under hocolims by the S^{i,j}. this is the colocalization of mspectra for which equivalences are detected by homotopy groups, rather than homotopy-sheaves-of-groups or whatever
so, motivically, usually to test if a map X --> Y is an equivalence, one needs to have an equivalence of all [[S^{i,j},X]] --> [[S^{i,j},Y]] -- i'm just randomly using [[-,-]] to denote the discrete-motivic-space of homotopy classes of maps. (i.e., the sectionwise \pi_0 of the hom-mspace)
fwiw, "jack's forms" refers to something arithmetic, rather than something structured: counting (incoherent) ring maps MU --> E_n which become conjugate in whatever relevant sense when passing to E_n (x)^ W(k bar)
so, probably not something hopkins--miller has something to say about, but also not something that requires much technology to answer
but then, if my mspectra are cellular, then i can just check on [S^{i,j},X] --> [S^{i,j},Y]
so: what is the analogous thing i get to forget if my equivariant spectra are cellular? i'd guess i can forget the maps involved in my mackey functors and just remember the values?
(I think that S^n ^ (G/H)_+ and S^V generate the same thick subcategories of the genuine-equivariant category, and so you get the same notions of weak equivalence)
so, mspectra are enriched in mspaces. in particular, we have a mspace map(S^{i,j},X). this has a 0-truncation, which is the discrete mspace [[S^{i,j},X]] -- one might call this the (i,j)th "homotopy motivic-group". these determine the equivalences of mspectra...whereas the usual [S^{i,j},X] (which are the global sections of [[-,-]]) are too weak
(the A^1-localization is a left localization, so you're basically shrinking the fibrant objects. so i'd guess that they are in particular fibrant as nisnevich presheaves-of-spaces -- i.e., they are sheaves)
i think you already gave the answer, which is what i expected: the analog of the motivic difference between[[S^{i,j},-]] and [S^{i,j},-] is equivariantly given by the difference between mackey functors and RO(G)-graded homotopy groups
and so a necessary condition -- at least on G-spectra for a trivial universe -- seems to be that these generate a full-rank submodule of Z^{# subgroups}
what's that? actually our most recent attempt was with correspondences where your backwards map needs to have a trivialization of its normal bundle
which, tantalizingly, suggests a more general statement for modules over various thom spectra (inasmuch as S = MFramed)
clark suggested that actually the difference between such spectral presheaves and actual motivic spectra might somehow be analogous to the difference between the sphere spectrum and its A-theory, but i can't remember anything more concrete than that
aha, that's not a bad idea. although it seems on the face of it like it might be stronger, like you're building in some thom iso's or something. a related guess would be only for trivialized principal G_m-torsors
it took me a minute to see why that was your guess
maybe i'm lying, G_m isn't the one-point compactification of anything that might deserve to be called the fiber of a vector bundle
@AaronMazel-Gee would you mind spelling out what this adams-atiyah/adams condition actually is? I don't necessarily need all of the details, but something more than "an Adams-Atiyah condition" would be really helpful.
@SeanTilson it's the thing that's necessary for a homotopy ring spectrum E to have a kunneth spectral sequence. i think the name is because it was first written down in adams's blue book, but he credits it to atiyah
namely: you need to be able to write E = colim E_\alpha for dualizable E_\alpha, such that: (a) the E_* DE_\alpha are projective E_*-modules, and (b) for any E-module spectrum M, the Kunneth map [DE_\alpha , M ] --> hom_{E_*-modules} ( E_*DE_\alpha, M_*)
given by taking a map DE_\alpha --f--> M to the composite E ^ DE_\alpha --> E ^ M --> M is an isomorphism
so i guess the key point, functionally speaking, is that E becomes "procorepresented" or something (probably that's the wrong word): for any X E_* X = [ S , E ^ X ] = colim [ S , E_\alpha ^ X ] = colim [ DE_\alpha , X ]
so somehow you're turning E-homology into something that smells like homotoyp
*homotopy
and this is essentially how goerss--hopkins reduce the E-homology obstruction theory to follow the story of Pi-algebra obstruction theory
