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

and a functor f:C->Vect with nice properties, there is a unique (and constructible, from the data) group G up to isomorphism so that (C,f) is equivalent to (G-modules,forget). You can also relax the properties a bit and replace "group" by "(quasitriangular) (quasi)Hopf algebra", etc. $\endgroup$4more comments