@DenisNardin Sorry, I'm a bit ignorant of theta rings but delta-rings are familiar to me. There is a larger framework of Lambda-rings that J. Borger champions I think convincingly, and in the realms of number theory and commutative algebra, Witt vectors are a primal indispensable example. What is the "theta" and how are these seem as the same?
@ManuelRivera okay, gotcha. certainly i believe that the ring spectrum $\Sigma^\infty \Omega X$ know about local systems on $X$, since they're just ring spectrum maps out of this thing. so your conjecture sounds good to me! what should be the inverse functor, though?
@dhy that is really cool, thanks for unpacking it. regarding "positive", i guess i meant: with respect to the standard definition of U(b), what do the simple perverse sheaves correspond to? but now i'm not sure that question is well-posed.
@S.carmeli i see. so it sounds like there are (at least) two hurdles: (1) group-completion, and (2) the difference between group-completion and the S-dot construction. still, that's really awesome -- and i agree that is feels tantalizingly similar to redshift...
@lemiller Theta rings are delta rings where you write the operation as $\theta$ instead of $\delta$. It's just a different name for the same thing. As it was mentioned here before, they arose in the study of power operations on K-theory
Hi all, do you know of any interesting example of a result in $\infty$-category theory whose proof goes as follows (or that could be proven in such fashion even though it's never been done so anywhere):
- the result can be proven for ordinary categories significantly more easily -the result can then be generalized to $\infty$-categories exploiting the fact that every $\infty$-cat is the localization of an ordinary category
@AaronMazel-Gee I guess that, for the A_n-quiver, you can explicitly represent them as follows. The root vectors of the unipotent correspond to the IC sheaf of quiver representations with a string of isomorphism of 1-dimensional spaces padded by 0-s from both sides. Then, the commutation relation come from the fact that, when you attach two such chains, the connecting map between then can be either 0 (one summand of the commutator) or arbitrary (the other summand).
Then you get as the commutator exactly the gluing of the chains (because we "cleared out" the IC of the locus where the connecting map of the chains is 0). But I'm really not sure about this description.
oh, sorry, what I said about the "clearance" in the commutator was nonsense, but I still think something like this should, maybe, not necessarily, work :-D
@EdoardoLanari I suspect there must be results about combinatorial model categories which can be read model-independently as results about presentable $\infty$-categories, which would fit the bill. I suspect that in many of these cases, a direct proof in the $\infty$-categorical setting is not so bad, but maybe nobody has bothered to write it down.
For example, right now I'm in the process of writing up a proof that if $C$ is a presentable $\infty$ category and $D \subseteq C$ is a full subcategory closed under limits and sufficiently filtered colimits, then $D$ is presentable, and an accessible localization of $C$. I suspect this could be deduced from combinatorial model category results, since the analogous result for locally presentable 1-categories is known. Instead I'm adapting the proof for 1-categories directly.
(It surprised me that I couldn't find this result for $\infty$-categories in the literature -- it seems the best that's known is that this is true if $D$ is already known to be accessible)
@AaronMazel-Gee Re: "with respect to the standard definition of U(b), what do the simple perverse sheaves correspond to?" It's a well-posed question, and the answer is very cool: They correspond to what is known as the "canonical basis" of U(b), which is hard to describe but turns out to be related to lots of other representation theory. So this Hall algebra description is actually useful for studying U(b)
@dhy so its not that the IC of a representation with 1-dimensional space in positions i-->j and 0 elsewhere and isomorphism between the 1-dimensional guys is the matrix coefficient in the case of A_n? its something else?
Hi all, I have a homotopy-coherence issue arising in a sheafy setting, and I'm trying to figure out whether \infty-categorical technology could help. Say I have a category C and a functor from C to schemes, c \mapsto X_c. Then it seems that there should be two bicartesian fibrations, classifying the action of f_!,f^! and f_, f^ on the derived categories D(Sh_{et}(X_c)) where f is a morphism of C. Is there a good place to look where these fibrations are constructed/discussed/studies?
@PhilTosteson one place to look at is Gaitsgory Rozenblium book "a study in derived algebraic geometry". Another, treating only the case where the maps are all proper, is Lury-Gaitsgory paper on weil conjecture for function fields.
@S.carmeli Thanks for the recommendations-- I remember that there was some controversy over the status of Gaitsgory--Rozenblyum, so I am a bit hesitant to invest a lot of time into trying to learn their approach. Though probably these constructions don't depend on the unproven assertions?
@PhilTosteson hmmm... well big deal of what they do is to take care also for base-change, and if you don't need it, then you can mennage I think.
The *-functor exist for general reasons (functoriality of sheaves in the site). Then to get the shriecks I guess you should post-compose with the monoidal duality of presentable k-linear categories (if you work with k-module sheaves e.g.). If im not mistaken this gives you the shrieks
Yeah, I guess ideally I would like to use proper base change and the projection formula freely-- but I imagine that's too much to ask for. This shriek fact you mention is surprising: Verdier duality intertwines shriek and *, but I don't know how to relate Verdier duality to an "intrinsic" operation (i.e. one that only knows about the category of sheaves and not the space that it is a cat of sheaves on).
@PhilTosteson it doesn't, but it gives you of a particular way to identify sheaves with their dual, in such a way that takes f^* to f_!, I think. So, using it pointwise, you can identify the objects and morphisms of the dual diagram with the other diagram you want, I think.
I say "I think" all the time because I'm far from confident about it.
more precisely, you have a diagram in the homotopy category of infty-categories with left adjoint functors, the one with the shriecks. You want to lift it to a diagram in Prl, or k-linear cats, e.t.c. So you take the dual of the diagram of the -s, which is already lifted. To *exhibit it as a lift of your original diagram, you use Verdier duality to produce the required identification (which is not unique so there's no problem of "intrinsicity" here).
