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!

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