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So I’m trying to draw some kind of connections here between a lot of really useful stuff that’s written on the nlab. I think it might even deserve its own page, which I would title “Descent Cohomology” (although maybe this is actually really quite trivial and doesn’t need its own page) but I think I need some help from you all to make it make sense in the “$\infty$” case. A lot of this has kind of been inspired by reading “Principal $\infty$-bundles - General Theory” by Urs, Thomas Nikolaus and Danny Stevenson, as well as stuff by Lurie, and a ton of other stuff for the discrete (and 2-categorical) case (Nuss and Wambst, Larry Breen, Knus and Ojanguren, SGA4, etc).
Given a cover in some ($\leq\infty$-) site $C$, $\phi:U\to X$, and some stack (or categorical bifibration?) $\mathcal{F}:C\to \infty-Cat$, I’d like to answer the question, for $M\in \mathcal{F}(X)$, what other $N\in\mathcal{F}(X)$ are there (up to equivalence) such that $\phi^\ast(M)\simeq \phi^\ast(N)$. This is basically asking for “twisted forms” of $M$, and if I’m not mistaken, this should be, at least theoretically, calculable as some kind of “cohomology” in some $\infty$-topos.
In the discrete case, one can compute such a thing by looking at $\check{H}^1(U\overset{\phi}\to X,Aut(\phi^\ast(M)))$, I believe. However, this also seems to compute principal $Aut(M)$-bundles for that cover as well. And in the nice case that the cover is “Galois” for some group $G$, this can be written down in terms of group cohomology of $G$ with coefficients in $Aut(M)$. There is machinery for doing something similar even in the more general scenario of Hopf-Galois extensions by some Hopf-algebra, explained very nicely in the paper by Nuss and Wambst: Non-Abelian Hopf Cohomology. What’s really nice about that scenario is that this same cohomology also classifies descent data for $\phi^\ast(M)$ (continuing with the notation from above). That is, it also classifies isomorphisms between the two different ways to pull back $\phi^\ast(M)$ to $U\times_ U$ (excuse me for skimping on the explanation here, the notation just gets unpleasant), or if we’re in the situation of monadic descent for some monad $T$, it classifies comodule-structures on $\phi^\ast(M)$ for the relevant comonad on the category of $T$-algebras (a relatively nice account of this is given in Mesablishvili’s On Descent Cohomology as well as Menini and Stefan’s Descent Theory and Amitsur Cohomology of Triples . So, this one single cohomology group computes a whole bunch of different things, which are all actually the same thing, and if we have a nice enough cover, we have even nicer ways of computing it.
So I guess my question is the following - Given all that we know about $\infty$-principal bundles (being computed by some $\mathbf{H}(X,BG)$), can we recognize this as some kind of descent cohomology, or higher Amitsur cohomology in the case of descent for either $\infty$-stacks or derived stacks? Now, a descent datum should be, instead of an isomorphism with a cocycle condition, a isomorphism with all higher cocycle conditions and bunch of cells gluing all of this stuff together (or in other words, the category of descent data is the limit of some simplicial $\infty$-category (or colimit and cosimplicial, depending on variance and so forth)). And so the “descent cohomology” in this scenario should be some higher, or derived, mapping space, but it should also depend on the choice of cover.
I’m trying to pick up the $\infty$ -categorical pieces as fast as I can, but I was just wondering if anyone has thought about this particular situation. I’d really really love to chat about it and try to get it ironed out.
Thanks!
Hi Jon,
you write:
In the discrete case, one can compute such a thing by looking at $\check{H}^1(U\to X,Aut(M))$, I believe.
Your paragraph before this sentence seemed to tell me that you wanted to look at $\check{H}^1(U\stackrel{\phi}{\to} X,Aut(\phi^\ast(M)))$ instead. Namely you’d be looking at objects on $X$ that have a local trivialization with respect to $\phi$ that is equivalent to $\phi^\ast(M)$ on $U$. That also seems to be what you say a few sentences afterwards:
What’s really nice about that scenario is that this same cohomology also classifies descent data for $\phi^\ast(M)$.
Would you agree? Or else maybe I misunderstood.
Apart from this, let me ask/remark:
First; one part of the question seems to be about the relation between what one might call “geometric descent” and dually “algebraic descent”, where in the first case one considers Cech nerves while in the second one considers their formally dual Sweedler corings/Hopf algebroids/Amitsur complexes. I.e. it seems you are after the $\infty$-version of the baby-case that is discussed at Sweedler coring – Geometric interpretation. Is that right?
Second; when you say you are after a notion of descent that explicitly depends on the choice of cover, then taken at face value that takes us out of the kind of the context as considered for instance in our “Principal $\infty$-bundles”. For there part of the whole point is that the choices of covers and their Cech cocycles etc. is all implicitly taken care of by the formalism and not part of the explicit picture. Or in other words, that Cech cohomology is a “model” or “presentation” of something more intrinsic, which does not depend on choices involved in the model.
Of course one could try to bring explicit choices of covers back into the intrinsic $\infty$-topos picture. I suppose one can say something here in terms of generalized relative cohomology. For instance if $\phi \colon U \to X$ is the given cover, then one can consider cocycles in the arrow $\infty$-topos $\mathbf{H}^{\Delta[1]}$ on $\phi$ with coefficients in $(\ast \to \mathbf{B}Aut(Q))$. One such cocycle is by definition a diagram of the form
$\array{ U &\to& \ast \\ \downarrow^{\mathrlap{\phi}} &\swArrow_{\simeq}& \downarrow \\ X &\to& \mathbf{B}Aut(Q) }$and this is, of course, equivalently a $Q$-fiber bundle on $X$ equipped with a trivialization of its pullback to $U$.
Would this be an “intrinsic” formulation of what you might be after (in parts, maybe)?
Hey Urs! Yeah, the first thing you said is definitely what I meant, so I just went ahead and edited it so that it would make more sense.
So - to your first statement; I would say that I definitely am, at the bottom of it all, interested in the affine case. In other words, I’d love to do this all in terms of monoidal objects in some category, and then possibly in a more general context of $\infty$-monads, but as I’m trying to develop with specific applications in mind, I really only care about so-called “Grothendieck descent” for modules over algebras. However, one can apply some suitable “Spec” functor to turn the Amitsur cohomology situation into a Cech nerve right? Anyway, that’s not super important. You’re right that I’m really only interested in the Sweedler coring situation, which I think Kathryn Hess did a pretty good job of describing (for simplicially enriched model categories, at least), just not up to the point I’m describing here. It is an interest of mine also to show that the category of co-cartesian modules for the Amitsur complex (as a co-cartesian ring) is in fact (Quillen, or homotopy, depending on the setting) equivalent to the category of corings over the Sweedler coring. Anyway… on to your second statement:
I think what you’re describing is exactly what I’m talking about!
The reason, I guess, that I’m interested in this is the following: typically one gets at so-called “twisted forms” for some object by taking the limit over all covers. That is, this gets us all forms which are locally isomorphic. However, I’m more interested in all possible descent data over some object. That is, if we had a map of rings $\phi:R\to S$, then there should be some kind of projection $Desc(\phi)\to SMod$, and I’d like to determine the fiber over a given module (even though in some cases it might be empty). In the case that I’m interested in an $S$-module which already carries at least one effective descent datum, I can compute this in terms of twisted forms.
Do you think your formulation, in terms of bundles in the arrow category, is computable?
Just to clarify the last thing I said:
Suppose I have a concrete sort of situation, e.g. I have a morphism of commutative (or even associative) ring spectra $R\to S$, and I want to answer this question in that setting. Then I should be able to choose cosimplicial models of everything I’m interested in (basically the Amitsur complex) and run some Bousfield Kan spectral sequence to get at the homotopy of the mapping space $[\phi,*\to BAut(Q)]$ whose $\pi_0$ is isomorphism classes of descent data on $Q$, where $Q$ is some $S$-module with an effective descent datum (which becomes the base point in the space whose homotopy I’m computing).
Augh, I made a mistake above. Indeed what I want to be computing is $H^1(\phi, Aut(M))$ rather than $H^1(\phi, Aut(\phi^\ast(M)))$, but the idea is the same.
Do you think your formulation, in terms of bundles in the arrow category, is computable?
So one thing one case say right away is the following trivial fact, which I mention for emphasis: if $\mathbf{H}$ is an $\infty$-topos with site $\mathcal{C}$ then the arrow topos $\mathbf{H}^{\Delta[1]}$ has site $\mathcal{C} \times \Delta[1]$. Therefore once you find that Cech cohomology on $\mathcal{C}$ “is computable” it should also be computable for $\mathcal{C} \times \Delta[1]$.
Now of course you want to pass to the dual algebraic formulation. Here I am not sure what I can say beyond the evident formal statements. I haven’t really looked at the articles that you link to in #1 yet.
Hey Urs,
Sorry, didn’t get to respond to this for a while. I guess the formulation I’m thinking of is the following:
Suppose I have a morphism of ring spectra (or ring objects in some suitable category $\mathcal{C}$) $f:R\to S$, and some ($\infty$) stack ${F}:\mathcal{C}\to Cat_\infty$, or something along these lines. Let’s also suppose that $M\in{F}(R)$. Then we can produce a space valued stack $hAut(M)$ which to $g:R\to S'$ associates the space $hAut(g_\ast(M)).$Then the hypothesis should be the following - The space of twisted forms for $M$, i.e. other objects $M'$ such that $f_\ast(M)\simeq f_\ast(M')$ is the totalization of the cosimplicial space $hAut(M)(R/S^\bullet)$ where by $R/S^\bullet$ I mean the Amitsur complex associated to the map $f:R\to S$. This, I feel, should be a result that holds quite generally. I’m currently trying to prove it. Moreover, if it’s true, then the standard computation of twisted forms of modules or algebras follows pretty immediately.
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