What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me)
Reconstruction of commutative schemes
Given a quasi compact and quasi separated commutative scheme $(X,O_{X})$ (actually, quasi compact is not necessary), we can reconstruct the scheme from $Qcoh(X)$ as category of quasi coherent sheaves on $(X,O_{X})$. This is Gabriel-Rosenberg theorem. Let me sketch the statement of this theorem which led to the question I want to ask:
The reconstruction can be taken as geometric realization of $Qcoh(X)$ (abelian category). Let $C_X$=$Qcoh(X)$. We define spectrum of $C_X$ and denote it by $Spec(X)$
(For example, if $C_X=R-mod$.where $R$ is commutative ring, then $Spec(X)$ coincides with prime spectrum $Spec(R)$). We can define Zariski topology on $Spec(X)$, we have open sets respect to Zariski topology. Then we have contravariant pseudo functor from category of Zariski open sets of the spectrum $Spec(X)$ to $Cat$,$U\rightarrow C_{X}/S_{U}$,where $U$ is Zariski open set of $Spec(X)$ and $S_U=\bigcap_{Q\in U}^{ }\hat{Q}$
(Note:$Spec(X)$ is a set of subcategories of $C_{X}$ satisfying some conditions, so here,$Q$ is subcategory belongs to open set $U$ and $\hat{Q}$=union of all topologizing subcategories of $C_{X}$ which do not contain $Q$.)
For each embedding:$V\rightarrow U$, we have correspondence localization functor: $C_{X}/S_{U}\rightarrow C_{X}/S_{V}$. Then we have fibered category over the Zariski topology of $Spec(X)$, so we have given a geometric realization of $Qcoh(X)$ as a stack of local category which means the fiber(stalk)at each point $Q$,
$colim_{Q\in U} C_{X}/S_{U}$=$C_{X}/\hat{Q}$ is a local category.
Zariski geometric center
Define a functor $O_{X}$:$Open(Spec(X))\rightarrow CRings$
$U|\rightarrow End(Id_{C_{X}/S_{U}})$
It is easy to show that $O_{X}$ is a presheaf of commutative rings on $Spec(X)$, then Zariski geometric center is defined as $(Spec(X),\hat{O_{X}})$,where $\hat{O_{X}}$ is associated sheaf of $O_{X}$.
Theorem:
Given X=$(\mathfrak{X},O_{\mathfrak{X}})$ a quasi compact and quasi separated commutative scheme. Then the scheme X is isomorphic to Zariski geometric center of $C_X=Qcoh(X)$. If we denote the fibered category mentioned above by $\mathfrak{F}_{X}$.
then center of each fiber (stalk) of $\mathfrak{F}_{X}$ recover the presheaf of commutative ring defined above and hence Zariski geometric center. Moreover, Catersion section of this fibered category is equivalent to $Qcoh(X)$ when $X$ is commutative scheme
Question
It is well known that for a compact group $G$, we can use Tanaka formalism to reconstruct this group from category of its representation $(Rep(G),\bigotimes _{k},Id)$. Is this reconstruction Morally the same as Gabriel-Rosenberg pattern reconstruction theorem?
From my perspective, I think $Qcoh(X)$ and category of group representations are very similar because $Qcoh(X)$ can be taken as "category of representation of scheme". On the otherhand, group scheme is a scheme compatible with group operations. Therefore, I think it should have united formalism to reconstruct both of them. Then I have following questions:
1 Is there a Tanaka formalism for reconstruction of general scheme$(X,O_{X})$ from $Qcoh(X)$?
2.Is there a Gabriel-Rosenberg pattern reconstruction (geometric realization of category of category of representations of group) to recover a group scheme? I think this formalism is natural (because the geometric realization of a category is just a stack of local categories and the center(defined as endomorphism of identical functor)of the fiber of this stack can recover the original scheme) and has good generality (can be extend to more general settings)
Maybe one can argue that these two reconstruction theorems live in different nature because reconstruction of group schemes require one to reconstruct the group operations which force one to go to monoidal categories (reconstruct co-algebra structure) while reconstruction of
non-group scheme doesn't (just need to reconstruct algebra structure).
However, If we stick to the commutative case. $Qcoh(X)$ has natural symmetric monoidal structure. Then, in this case, I think this argue goes away.
More Concern
I just look at P.Balmer's paper on reconstruction from derived category, it seems that he also used Gabriel-Rosenberg pattern in triangulated categories: He defined spectrum of triangulated category as direct imitation of prime ideals of commutative ring. Then, he used triangulated version of geometric realization (geometric realization of triangulated category as a stack of local triangulated category) hence, a fibered category arose, then take the center of the fiber at open sets of spectrum of triangulated categories to recove the presheaf (hence sheaf) of commutative rings. This triangulated version of geometric realization is also mentioned in Rosenberg's lecture notes Topics in Noncommutative algebraic geometry,homologial algebra and K-theory page 43-44, it also discuss the relation with Balmer's construction. But in Balmer's consideration, there is tensor structure in his derived category
Therefore, the further question is:
Is there a triangulated version of Tannaka Formalism which can recover P.Balmer's reconstruction theorem. ? I heard from some experts in derived algebraic geometry that Jacob Lurie developed derived version of Tannaka formalism. I know there are many experts in DAG on this site. I wonder whether one can answer this question.
In fact, I still have some concern about Bondal-Orlov reconstruction theorem but I think it seems that I should not ask too many questions at one time.So, I stop here.All the related and unrelated comments are welcome
Thanks in advance!
Reconstruction theorem in nLab reconstruction theorem
EDIT: A nice answer provided by Ben-Zvi for related questions
Best Answer
The Tannakian reconstruction for schemes and more generally, geometric stacks from QCoh(X) with its tensor structure (due to Jacob Lurie) is explained in my answer to Tannakian Formalism . One can (and one has) try to extend this to the derived setting, where one ought to get a statement much stronger than Balmer's reconstruction theorem. First of all one needs to replace triangulated categories, which are too coarse to work with effectively, with symmetric monoidal $(\infty,1)$-categories (or symm. monoidal dg categories say in characteristic zero).
There's an obvious functor from such objects C to derived stacks Spec C given by Tannakian reconstruction -- it's defined just as in the abelian case (see my answer above). Namely we build Spec C's functor of points, as a functor from derived rings to spaces or simplicial sets: Spec C(R) = the $\infty$-groupoid of tensor functors from C to R-modules. In the case that C is the infinity-categorical form of QCoh (X) for X a scheme this recovers X, giving a result refining Balmer's (this was worked out by Nadler, Francis and myself, and many others I think -- in particular Toen and Lurie understood this long ago). I feel strongly this is a better point of view than trying to define a locally ringed space from a triangulated category, and has more of a chance to extend to stacks. In particular C tautologically sheafifies (as a sheaf of symmetric monoidal $\infty$-categories) over X.
Things get much more interesting though in the case that C=Qcoh X for X a stack (with affine diagonal, or everything fails immediately) -- eg for C=Reps(G)=QCoh (BG), the setting of usual Tannakian formalism. There one quickly learns that the above naive picture is false. Namely take G to be just the multiplicative group $G_m$. Then there are a lot of interesting fiber functors on the derived Rep G = complexes of graded vector spaces that are not given by points of BG_m (ie are not the usual "forgetful" fiber functor) --- namely we can send the defining one-dim rep of $G_m$ not to the one-dim vector space in degree zero, but in any degree we want (ie apply shift) - since this is a generator it determines the rest of the fiber functor. This proves that a naive Tannakian theorem fails in the derived setting, without imposing some "connectivity" hypotheses (ie without having t-structures around). I believe a student of Toen's is writing a thesis on this subject, but I don't know the precise statements (and so won't "out" his results).
Of course it's a little disappointing to need BOTH a tensor structure and a t-structure, since morally either one should be enough for reconstruction (having a t-structure means we can recover the abelian category of quasicoherent sheaves, which by Rosenberg recovers X already in the case of schemes without need for tensor structure). But for general stacks I don't know of a better answer.