@Spice the Bird: On the bright side, Shaferevich conjectured that the the Galois group ramified at only finitely many primes is topologically finitely generated. I don't know if it's proved or not.

@Eitan: No -- a Grothendieck abelian category is like the category of all modules over a ring. It has infinite direct sums, filtered colimits preserve monomorphisms, and it satisfies a (very mild) set-theoretic condition. But it doesn't really matter -- all that's important is the existence of enough injectives (which the Grothendieck condition implies). So the dual thing is satisfied if there are enough projectives.

@Fernando: I don't see how to deduce that the adjunction always exists from the theorem you cited. The hypotheses are not dissimilar from what I wrote in my post above. (Note the author's use of the adjective "absolute").

A hypothesis I've found useful before with this question is that if the categories are Grothendieck abelian categories and $F$ is exact, then $LF$ and $RG$ are adjoint on bounded below derived categories. This follows immediately because $G$ preserves injective objects.

With that said, I don't see any problem formulating the universal property for blow-ups in the derived setting -- the notion of Cartier divisor is no problem. However, the usual proof of representability doesn't go through, and it may take some fiddling with deformation theory to show that it is representable.

I'm not sure what to say -- it's hard to argue why a notion does not exist. If you look in Sections 4 and 5 of DAG XII, you can see Lurie avoiding the notion of $n$th power of an ideal, but I don't see anywhere that he explicitly says that it doesn't exist or why. The essential notion which is missing here is image of an ideal under a map, which doesn't exist since image isn't a notion in a stable $\infty$-category. By the way, even the notion of ideal is fraught in the derived setting: e.g., the only invariant notion of quotient I know of is a map $A\to B$ which is surjective on $\pi_0$.

I've heard from experts that the answer is no: there is not a good notion of powers of an ideal. (Say, invariant under quasi-isomorphism, or definable in Lurie's framework). So there's no $n$th infinitesimal neighborhoods in DAG, only formal completions.

By the way: $\mathbb{G}_m(\mathbb{C}((t)))$ shouldn't be thought of as an affine space minus a point -- as an ind-scheme it is $\mathbb{Z}$ times an infinite dimensional affine space. (Which isn't surprising homotopically since $\Omega S^1=\mathbb{Z}$).

The case of a more general group won't be so simple: the action of the principal adelic points is more subtle for a non-abelian group. However, Gaitsgory and Lurie have announced a computation of the Atiyah-Bott formula along the lines of what you ask. The argument is by realizing the homology of Bun_G as the factorization homology of the homology of the affine Grassmannian. The homology of the affine Grassmannian is easy to compute (as in your case above) and then the corresponding factorization homology turns out to be quite simple. See Gaitsgoryarxiv.org/pdf/1108.1741 for some more details.

@Akhil: Maybe I misunderstand you, but there's no reason a priori why that 2-limit should have anything to do with the homotopy category of the homotopy limit. (Okay, there's a functor in one direction, but it has no a priori reason to be an equivalence -- one has to use something about the particular situation). As for a construction of the homotopy limit -- I suggest you try to write it down for yourself as an exercise.

@Tommaso: Thanks for the link. In fact, when I first asked around about this, I was referred to fherzig's post! :) @Emerton: I am aware of Serre-Tate theory. However, I don't find it satisfying because I'm not sure how to construct a category $\mathscr{C}$ fitting the above.

@algchen: The associated gradeds can indeed be different, but in both cases the set-theoretic support of the modules is the zero section of the cotangent bundle. (Note that in the VHS case the action of vector fields is nilpotent on the associated graded.)

@Akhil: I was imagining that a person would construct the derived category of $\ell$-adic sheaves the higher categorical way, then show it has a ``perverse" $t$-structure, show that the heart coincides with the usual perverse heart, and then redo Beilinson's construction to compute Exts. I wouldn't say it's any kind of refinement, but rather a way to show that the smart construction coincides with the usual construction (which is out of laziness: then one doesn't have to reprove all the theorems).

You should add the adjective "quasi-triangular" in front of "Hopf algebra." This condition exactly ensures that the category of representations is braided monoidal.

@Akhil: Perhaps one would prove that (or, say, the $\mathbb{Q}_{\ell}$-analogue) by showing both sides are the derived categories of perverse sheaves, the classical definition having been done by Beilinson and the $\infty$-categorical version amenable to a similar treatment. But it would definitely be nice to have it written up. The limit Deligne takes appears (to me) to work only out of "good luck," (since it's not a real operation in the world of triangulated categories) whereas the homotopy limit is much more natural.