@AaronMazel-Gee Here is an attempt at an explanation (I am a bit tired rn so there may be some moments of incoherence): My v is a vector encoding the dimensions of the vector spaces in the quiver representation, and I mean -v^TAv in the sense of linear algebra. More precisely:

Let the vertices in my quiver be labeled 1,2,...,n. Say I require my quiver representation to assign to vertex i a vector space of dimension a_i. Once we fix these dimensions, we can construct a stack of quiver representations as follows. Let X be the product over all quiver arrows i->j of Hom(C^{a_i},C^{a_j}). This is the moduli of "framed" representations - to get the moduli stack we quotient by the product of the Aut(C^{a_i}). The dimension of this moduli stack is:

sum_arrows a_ia_j - sum_vertices a_i^2

taking v to be the vector with components a_i, this is (up to a factor of 2) -v^tAv for A equal to the Cartan matrix

and so A positive definite is exactly when this space has dimension <= 0, and so where you have a chance of having finitely many representations of each dimension, which in turn is necessary to have a chance of having finitely many indecomposable representations in total

(So everything stated so far is very concrete - I build a "moduli of quiver representations" in a naive way, and then I do some linear algebra to calculate when its dimension is nonpositive)

The Hopf algebra structure is a bit harder to describe. IIRC it goes like this: Let Rep_v be the space of representations with dimension vector v. Then for the multiplication operator you want a map K^0(Rep_v) x K^0(Rep_w) -> K^0(Rep_v+w). This comes from a specific correspondence X; namely the moduli of exact sequences of quiver representations 0->A->B->C->0, where dim A=v and dim C=w. So you get maps X->Rep_v x Rep_w and X->Rep_{v+w}, and the desired multiplication map is the composition

pullback K^0(Rep_v)xK^0(Rep_w)->K^0(Rep X) and pushforward K^0(Rep X)->K^0(Rep_{v+w})

comultiplication comes from the same correspondance, but pull-push in the opposite order; and the antipode is IIRC Verdier duality?

Oops, I forgot to write that by K^0 of a space I actually mean K^0 of the space of perverse/constructible sheaves on that space

(The original motivation for this construction was that this gives you a very natural basis of U(b) (and by extension U(g)): Take the basis defined by simple perverse sheaves!) I'm a bit unsure which meaning of "positive" elements of U(b) you are referring to

@AaronMazel-Gee we are mostly interested in the stable settings. But you can think of higher commutative monoids to צ^[n] (the universal stable n-semiadditive category) as the relation of spaces and spectra. Even if eventually you care only about spectra, you still want to use unstable methods sometimes. One thing which we want to understand better, mostly in relation with K-theoretic questions (since its a generalized group completion), is the stabilization functor CMon_n--> צ^[n].

regarding the red-shift conjecture. What we know, as shown in the paper, is that the monoid of object of an ambidextruous category of height n carries a structure of a higher commutative monoid of height <= n+1. But K-theory involves also group completion, which in principle might destroy the height, and even the higher commutative monoid structure.

Put differently, if you K(j) localize K(R), you have a structure of a higher commutative monoid on this object (as every object of Sp_{K(j)}), and we want to know its wrong way maps are related with the formation of colimits on R-valued local systems (i.e. the "internal" and "external" push forwards are related).

Regarding the comparison of height, we thought for a long time that elements of height n in צ are T(n)-local. You can show that they have to be T(j)-asyclic for j!=n, but what you don't know is that they don't have "a-telescopic" part, i.e. one which is T(j)-acyclic for every j. But in fact, there's a counter example to the latter, and there's an object of צ of height 1 in the semiadditive sense which is not T(1)-local, it has a "contribution at infinity".

*"non-telescopic"

Finally, I know its obvious to you, but I still feel the need to make it clear that everything I told is a joint work of our ambigroup (Tomer Schlank Lior Yanovski and I) and im really only the representative that happened to be here among us.

oh, sorry, my mistake, i meant j>n in the T(j)-locality statement.

Hello everyone, I have some questions about elliptic cohomology. In Lurie's survey paper, he seems to suggest that the multiplicative group G_m can be defined over the sphere spectrum, but the additive group G_a cannot, so he only talks about it over Z. This surprises me because I thought that we can take $\Sigma^\infty_+ \mathbb N$ over the sphere spectrum. In David Gepner's thesis he also talks about the "strict additive group" as the thing this corepresents. Is this a mistake then?

@Gasterbiter I think the problem is defining the Hopf algebra structure on $Σ^∞_+N$ (since it cannot come from a monoid structure in spaces), but I haven't thought about it deeply. Intuitively I would imagine it can be defined though, by showing it corepresents the functor Map_{E_∞}(N,Ω^∞-) which is group-valued

@DenisNardin Thanks! Ok, I see, so it is hard to define a monoid structure on $Σ^∞_+N$. Is it easy to why Map_{E_∞}(N,Ω^∞-) is group valued though?

@Gasterbiter Intuitively I'd say yes (since E_∞-groups is an additive ∞-category), but again I haven't thought things through

Thank you, I'll think about that. I wonder why Lurie doesn't mention this then. But maybe he only cares about Q anyway.

@DenisNardin Is it a problem Ω^∞X isn't group like? (with respect to the multiplicative structure) (sorry to keep bothering you, feel free to ignore)

@Gasterbiter I'm using the additive structure -- I thought we wanted to get G_a?

Ah ok, sorry

@DenisNardin Now that I think of it I'm not sure this corepresentability result is true though, @Gasterbiter

I guess the question is whether that functor ("the strict additive group") is actually representable, and it doesn't feel at all obvious how to go around it because you want a ring where the additive structure is strict but the multiplicative structure isn't

It doesn't follow from adjunction?

No, remember that we want maps of E_∞-spaces. not just maps of spaces

Hmm, Gepner states this result in his thesis (www-users.math.umn.edu/~tlawson/tmf-mirror/… Proposition 14) but doesn't prove

I'm very skeptical of that proposition now. I don't have time to dig deeper but it feels wrong: Σ^∞_+N has strict multiplicative structure but lax additive structure, which is the exact opposite of what we want

@DenisNardin ok, thanks

Where does the lax additive structure come from btw?

@Gasterbiter Well, that's because it's a spectrum, its Ω^∞ always has a canonical additive E_∞-structure

@DenisNardin oh, duh

isn't the adjunction \Sigma^\infty_+, \Omega^\infty monoidal? (in the sense that the right adjoint is moniodal and the left adjoint is (op?)lax monoidal)

And if so shouldnt that imply that there is an induced adjunction from E_infinity spaces to E_infinity-algebras? Maybe that's what Gepner is using @DenisNardin?

You want to be really really careful about what monoidal structures you put there. FWIW I believe in the existence of an adjunction (because Ω^∞ preserves all limits & filtered colimits), but I don't believe that Σ^∞_+ is the left adjoint of this particular pair

In particular notice that Ω^∞ is only lax monoidal for the structures we are considering (since Ω^∞(A⊗B)!=Ω^∞A×Ω^∞B)

Oh whoops, I guess I meant Σ^∞_+ is symmetric monoidal so the right adjoint is (op)lax monoidal. Or isn't it?

It depends on what monoidal structures you're considering..

The argument you describe gives Ω^∞ the lax symmetric monoidal structure that endows Ω^∞E with the multiplicative E_∞-structure for E a ring spectrum. But that's not what you want!

You want to consider Ω^∞ as a symmetric monoidal functor for the product on both sides. But then this endows Σ^∞_+ only with an oplax symmetric monoidal structure and that's not enough for what you're trying to do (lift the adjunction at the level of algebras)

@DenisNardin now I understand. Wow this is confusing

So essentially what your saying is that Spec Σ^∞_+N and G_a = Map_{E_∞}(N,Ω^∞-) are two different functors CAlg -> CMon(S)^gp, while multiplicatively, the analogous two things are equal

So that's why you don't really have an additive group "scheme" over S...

Does there exist a presentable, stable $\infty$-category for which cohomological Brown representability fails (ie. not every cohomology theory is representable)?

I think Neeman's generalization of the classical argument shows Brown representability always holds in the compactly generated case, but I'm not sure what's known beyond.

@PiotrPstrągowski Yeah, the compactly generated case follows, e.g., from HA.1.4.1.10 (.which is in a slightly more general setting). I suspect that if your category is not cg there will be counterexamples though

That's what I thought, too. However, isn't any stable, presentable $C$ an exact localization of a compactly generated $D$, ie. we have an $L: D \rightarrow C$ with a fully faithful right adjoint? Then any cohomology theory $H$ induces one on $D$ via the formula $H \circ L$, which is necessarily now representable in $D$. But this object must be $L$-local by looking at the cohomology theory it represents and so actually it is an object of $C$.

I do suspect this is a little bit too naive, though, but I can't quite pinpoint what goes wrong.

@PiotrPstrągowski is it really true that every presentable stable thing is an exact localization of a compactly generated one?

I would've thought you'd need to replace "compact" with "$\kappa$-compact" for some $\kappa$

@DylanWilson It is certainly true that every presentable $C$ is the localization of a cg one $D$ (in fact of a category of presheaves), and I think that by stabilizing (i.e. taking spectral presheaves) you can force $D$ to be stable if $C$ is

@DenisNardin an exact localization? I would've thought you could only be an accessible localization of presheaves

@DylanWilson Ah I missed the word "exact"... but why do we need it for the above argument actually?

The hypothesis for Brown representability are satisfied whenever L is a left adjoint, isn't it?

hmm... something seems fishy like we might be trading compactness for $\kappa$-compactness somewhere, but I can't see where yet...

alright, let's accept that part for now, and move on to the claim that the representing object is local.

how's that go?

hmm... yeah that also seems fine.

alright, let's go back to that claim from before: every presentable, stable thing is a localization of spectral presheaves. is that true?

@Gasterbiter so for an E_infty space X, a map ℕ -> X is sometimes called a "strictly commutative" element of X. strictly commutative elements of X are closed under the product. the adjunction between E_infty spaces and commutative ring spectra says that a map of E_infty rings Σ^∞_+ ℕ -> A is the same as a strictly commutative element of (Ω^∞ A)_⊗

meaning, Ω^∞ A under multiplication. these are sometimes called strictly multiplicative elements. the issue that comes up is that while strictly multiplicative elements are closed under multiplication, they are not closed under addition

@DylanWilson: Naively: write C = Ind_kappa(C_0), where C_0 is the small stable category of kappa-compact objects. Then SP(C_0) -> C [SP = spectral presheaves] preserves colimits and kappa-compact objects, hence the right adjoint preserves kappa-filtered colimits, which I think implies that C -> SP(C_0) is fully faithful.

yeah... I guess this is true.

well, I tried to poke holes, Piotr, but it didn't work! so maybe the argument is just correct? ... weird...

this is basically what Denis said about the lax addition being a problem.

(and I guess HA.1.4.4.9 if we need some extra confidence about that localization fact)

in particular, if we take the generator of R = Σ^∞_+ ℕ and call it "t", then in R⊗R the element t⊗1 + 1⊗t isn't strictly multiplicative. this means that we can't get the coalgebra structure

the proofs i know of this use something called the mixed Cartan formula that expresses how "multiplicative" commutativity interacts with the distributive law

One proof is to K(1)-localize, so that pi_0 will be a theta-ring. Strictly commutative elements x satisfy theta(x) = x, which fails for t⊗1 + 1⊗t.

@DylanWilson, @DenisNardin I just added "exact" because that's an important condition in other contexts, but of course this is true for any left/right adjoint in the stable setting, because left exactness implies right exactness and vice versa.

@PiotrPstrągowski Right, then I cannot see a mistake in your argument :)

I agree this is strange, I always thought compact generation was necessary. On the other hand, I can't find any counterexamples in the presentable case either.

A lot on the subject was written by Neeman, but from what I can see, the counterexamples use rather exotic triangulated categories (that are probably not homotopy categories of anything presentable), or standard stuff (such as derived categories of rings), but for a different question (Neeman et al. show that not every cohomological functor on perfect complexes is representable, but that's probably different from cohomology theories on all of D(R) by some lim^1-terms.)

Oops, by theta(x) = x I guess I mean theta(x) = 0.

@TylerLawson @WilliamBalderrama Thanks! This helps me understand better what Dennis was saying.

@WilliamBalderrama This looks like a useful obstruction - where can I read about "theta rings?"

for K(1)-local homotopy in particular, mike hopkins has a fun document called "K(1)-local E∞-ring spectra", which is printed in the tmf book. (as a warning, there's an operation theta which exists in the homotopy of K(n)-local E_n-algebras for any n, and it always satisfies some equation relating theta(x+y) to theta(x) and theta(y). but theta(xy) generally does not satisfy some equation involving theta(x) and theta(y) if the height is >1. idk how relevant this is to the discussion tho)

@gasterbiter The paper @skd gives is probably the standard reference. I learned a lot of this stuff by figuring out what in Atiyah's "Power operations in K-theory" can be made to work for ordinary p-adic K-theory.

Standard reference for their relations to K(1)-local E-infinity rings I mean. I don't know offhand a good reference for just the plain algebra of theta rings.

Fun fact: theta rings are perhaps better known to number theorists as delta-rings (as those used to define prismatic cohomology). I do wonder if there's a connection on the same algebraic structure popping out in unrelated parts of math...

for theta/delta-rings in particular, there's a lot of good stuff in arxiv.org/pdf/1905.08229.pdf. one consequence of some of the stuff they discuss which i find somewhat interesting is the following: if you take the free E_oo-KU_p-algebra on one generator x, and kill the element psi^p(x) in an E_oo-way, you get an E_oo-KU_p-algebra which has divided powers on x (on homotopy). (basically, you get to now write down x^p/p, and that's basically all you need to get all divided powers on x)

this is supposed to be a pd-analogue of a result of hopkins-lurie at height 1, which says that if you take the free E_oo-KU_p-algebra on one generator x, and kill theta(x) in an E_oo-way, then you get the flat affine line KU_p[x] (see the first lecture in math.uchicago.edu/~amathew/notes_thursday.pdf). hopkins and lurie's result admits a generalization to higher heights, so it might be possible to generalize this pd result too. idk how interesting this is though

A similar fun presentation (explained in arxiv.org/abs/1406.5620) that I wish I understood better is that if you instead set psi(x) = x, you get (KU_p)_* KU_p.

@skd If the operations you refer to are the one's I'm thinking of, then they exist for K(n) local commutative rings, and you don't need the E_n-module structure for them. you talk about the ones given by tracing over BSigma_p the absolute p-power operation, like in mikes note for height 1?