@BrunoStonek There's a proof of some Fontaine's conjectures by Bhatt and Beilinson using derived de Rham cohomology. These theorems make comparisons isomorphisms between de Rham (crystalline) and p-adic étale cohomology after tensoring by a ring of periods.math.uchicago.edu/~drinfeld/p-adic_periods/… . There's also a proof of Artin reciprocity for function fields using derived stacks by a student of Gaitsgory, but I can't find it now.

Does anyone here know some elementary references for topological Hoschild cohomology and topological cyclic cohomology? Also is there any definition of THH that goes along the lines of nlab page definition, which is HH (X):= O(LX)?

@user40276 thanks for the references! About topological Hochschild cohomology: I don't know what you're looking for exactly, but there are some things in Lazarev's "Cohomology theories for highly structured ring spectra"

see also Basterra-Mandell, homology and cohomology of E_infty ring spectra, Corollary 8.2 for an interesting result using topological Hochschild cohomology

@BrunoStonek I think arxiv.org/abs/1710.05902 is an amazing application, to a question which is purely classical

@DenisNardin i agree that that's a nice way of repackaging it, but i wouldn't say that's "the point". i'm more just curious about the conditions on a (pre?)stable $\infty$-category that make this true. though sure, i'll happily bite: what's the lawvere theory repackaging that you have in mind?

Well, you can describe connective spectra as algebras in spaces for a certain Lawvere theory (given by the subcategory of Sp spanned by finite sums of the sphere spectrum), and this Lawvere theory is symmetric monoidal (inherited from that of Sp) so you can equip its algebra with the Day convolution symmetric monoidal structure. Since sets are a suboperad of spaces (in fact they are closed under the cartesian products!) it is more or less obvious that algebras in sets form a suboperad [..]

@user40276

[..] of the algebras in spaces. A nice thing is that this Lawvere theory has a completely combinatorial description: it is the objectwise group-completion of the span category of finite sets

sorry, i'm getting confused. what's a "symmetric monoidal" lawvere theory, and where did (sub)operads get involved?

iirc a lawvere theory is a category whose equivalence classes of objects are in bijection with 1,2,3,..., some (co?)product condition, etc. etc.

Well, a Lawvere theory is a category with finite products (plus some other conditions I don't care about), I just meant that it also has a symmetric monoidal structure that is compatible with the products

so here products are direct sums, and you're noting the extra structure coming from smash product?

i guess specifically n*m ~~~> nm.... or something like that. is that would you mean by "compatible with the products"?

i mean that the tensor products distributes the direct sums

I guess my observation is that Fun^×(L,-) sends suboperads to suboperads for a sufficiently nice L

okay right, i think that's equivalent then. in other words, the only way for a lawvere theory to be compatibly symmetric monoidal is for the underlying function on the set of objects to be $(m,n) \mapsto mn$

And since sets are evidently a suboperad of spaces, abelian groups are a suboperad of connective spectra

Possibly, but the reason I find this description interesting is that you can apply it verbatim to "many objects Lavwere theories", like the one that describes connective G-spectra

that's neat, it hadn't occurred to me that you can day-convolve models for a lawvere theory

@DenisNardin ah, even better

I mean, it doesn't work always, but when it does it is a cool trick :)

okay but so sorry, i'm still slightly confused. you're talking about operads, but usually i only hear people talk about representing a lawvere theory into a cartesian symmetric monoidal category. are you implying that you can ask for models in an arbitrary symmetric monoidal category? (i don't see why not, just from the shadow-definition i'm vaguely recalling)

No, I'm just considering models in spaces

I'm simply implying you can use the symmetric monoidal structure on your Lawvere theory to get a symmetric monoidal structure in your category of models

and that this sends subcategories closed under cartesian product to suboperads

ohhh of course, for the day convolution you need a s.m. structure on both source and target

are you saying "operad" as shorthand for "s.m. category", because it's more general?

No, a suboperad is not the same thing as a sm category

I mean, both objects are sm categories but the condition of being a suboperad is weaker than being closed under tensor product

It means that Map(⨂_i X_i,Y)→Map(⨂_i F(X_i),F(Y)) is an equivalence

okay gotcha, so you're using Spaces as a s.m. category (maybe even cartesian) but the point is that Sets can be a suboperad without the inclusion being strong s.m.

Well, the inclusion of Sets is strong sm

okay, now i see what you

're saying

(of course, Sets --> Spaces does happen to be strong s.m.)

But the inclusion of abelian groups in connective spectra is not

(the point is that suboperad is all you need for the functor on algebras to be fully faithful)

okay, and then finally: you're saying that $Models(\{S^{\oplus \bullet}\},Sets^\times) \simeq A(Fin)$ (as opposed to $A^{eff}$)

@DenisNardin right, that was where we started

No, $\{S^{⊕\bullet}\}=A(Fin)$

$Models(\{S^{\oplus \bullet}\},Sets^\times) \simeq Ab$

ohhh, like a many-object BPQ

Well, it's equivalent to BPQ

sure, i'd believe that

okayyyy and then you're representing this thing into $Sets^\times$ and/or $Spaces^\times$

yes, and the point is that this operation sends inclusion of categories closed under products to suboperads (this is pretty much immediate from the definition of Day convolution if you look at it)

okay, so you're getting back the originally assertion i mentioned, rephrased in terms of operads for Sp and Ab, but coming from a totally different direction.... albeit using the input of recognizing certain categories of models as connective commutative ring spectra and commutative rings, respectively

Well, algebras in certain categories of models

err...right, thanks

(you can also do another proof by representing connective E_∞-algebras as models for a different Lawvere theory, although proving the combinatorial description is more involved)

ah, so then it's one thing to say "hey, i just got back Sp and/or Ab" but then also you need to identify this new symmetric monoidal structure as the one you want it to be

There aren't that many symmetric monoidal structures on Sp that commute with colimits separately in each variable...

(that is, the space of such is contractible)

sure sure

i'm just trying to identify the ingredients that are going into this argument, i find that helpful when trying to make some mental space for it

okay, this is cool. thanks!

@BrunoStonek The solution to Weibel's conjecture employs crucially the notion of derived blowups: arxiv.org/abs/1611.08466

@AaronMazel-Gee Tobi and I learned that from Tyler Lawson, and we included his argument as Lemma 5.5 in our Thom spectra paper. (It's the same as the argument you sketched.)

@BrunoStonek Thanks for the references. I think that I was looking for homology instead of cohomology. I confused negative cyclic homology with cyclic cohomology. About the reference that was missing in my previous comment it's arxiv.org/pdf/1710.02892.pdf . Now that I see it again it's not derived, it's just stacky though.

@AaronMazel-Gee Thanks for the reference. Your paper clarified some of my confusions. Indeed, I have confused negative cyclic homology with cyclic cohomology. Now about that "perfect stack", do you mean something like commuting with limits, more precisely the limit O (LX) = S^1 . O(X) ?

By your paper, it seems that THH (A) is simply given by the realization of the usual bar complex A \wedge A ... \wedge A (the wedge is over S here) by using a similar argument as the one given in the nlab (by the definition HH(X) := O(LX), they conclude that HH (X) for X = Spec (A) can be computed from the usual bar complex). However in math.mit.edu/~nrozen/juvitop/dundas.pdf pag. 125, it's said that this naive definition is incorrect (the corrected one is defined in pag.126)