in lurie’s elliptic cohomology survey he says that it is easier to lift a sheaf of multiplicative cohomology theories to E-infinity ring spectra than it is to lift from cohomology theories to spectra. If I use that there are no phantom maps between even-graded landweber exact cohomology theories, then it seems like I get a B-sheaf (presheaf?) on the affine étale site. What goes wrong with this picture exactly if I then want to get a sheaf from this?

@DenisNardin What happens if you look at the algebraic k-theory of vector spaces over $\mathbb{Q}_p$?

@CPM Using your method you get a diagram in the homotopy category $\textrm{hSp}$. In order to get a sheaf you need to get a diagram in the full category $\textrm{Sp}$ (this is indipendent of the model you use to describe spectra, you just need to be able to define homotopy limits)

@SeanTilson The $\pi_0$ is wrong. Why should this be a sensible model?

@DenisNardin I wasn't suggesting it as a solution, it is just my first guess at something in a relevant direction and I was curious about how it failed to be correct. I guess anything built out of vector spaces will have the wrong pi_0. Have you looked at the fiber sequences that Antieau-Barthel-Gepner right down?

Not that they contain a solution, but that I don't see how one would get pi_0 right, but maybe working with torsion modules you can get something. These are wild stabs in an undeserving darkness.

Which one? There are a lot of fiber sequences in that paper :)

Yeah, in any case we need some kind of objects whose dimension is naturally a $p$-adic integer, so I was hoping something more like a graded vector space modelling the $p$-adic expansion in $\mathbb{Z}_p$

I thought maybe p-torsion modules could be indexed by something like the sequence of dimensions in the associated graded of the natural filtration by powers of p. But then this seems like it could run into some Eilenberg swindle-ish thing by not being f.g.

Hmm that's interesting

I have no reason to believe it is a good idea, just seemed like a natural guess.

to me, but maybe it is wrong for obvious reasons.

How does one get $\pi_0(K\mathcal{C})=\mathbb{Q}$?

Where $\mathcal{C}$ is some symmetric monoidal category.

I don't know. Do you know of any such examples?

I mean, we could cheat and take $\mathbb{Q}$ as a discrete category but that's obviously not interesting

Well, lets not cheat.

Maybe @SaulGlasman knows.

I don't know any nice examples where K_0 = Q

I was about to say I don't even know any nice examples where K_0 is finite but actually I think Tate spectra often have finite K_0 so maybe that's a good starting point for trying to construct things with complete K_0

IIRC i think khovanov once said he already doesn't know a nice monoidal (maybe also triangulated) thing whose K_0 is Z[1/2]

i think I asked to get the p-local sphere on here a while back and Clark said he thinks he can get the mod p and p-complete sphere because he can build multiplication by p after doing the S-dot construction? I'm probably saying this wrong, but I can try to find it in the transcript later

but if that's true then presumably a similar thing will build mod p ku

Mac Lane makes the observation that the bar construction on a group (Milgram model) "can be defined conceptually (no d_i, s_i) using the fact that the monoid $2\to 1 \leftarrow 0$ in $\Delta$ is universal". I have no idea what he's talking about

@lentic I assume that he's using that $\Delta$ is the free monoidal category equipped with a monoid. That is if $C$ is a monoidal category then monoidal functors $\Delta\to C$ are uniquely determined by the image of $[1]$. Here the monoidal structure on $\Delta$ is juxtaposition ("ordinal sum")

hmm, so your Delta is the one giving -augmented- simplicial objects, I gather

Hmm... Yeah I suppose I should have used $\textrm{Ord}$ instead of $\Delta$

that's cool. I still don't see how that helps towards defining the bar construction "conceptually", though...

damn, yes I do, I was distracted

Is the theory of quasicategories, and Lurie's work therein, the only place that $\infty$-operads have really been developed? Does anyone have a method for talking about, say, multiplicative or algebraic structure in complete Segal spaces?

@JonBeardsley Didn't Hackney, Robertson, and Yau write a book on such things? (another approach to infty operads.)

I am probably wrong.

@JonBeardsley I wouldn't call it completely disjoint from the theory of quasicategories, but the theory of operator categories allows you to talk about monoidal structures for complete Segal spaces