I think I should elaborate a bit. What I am asking is the definition of spectrum of a category as a stack in functor view of points.

In noncommutative algebraic geometry. We define spectrum of an abelian category or triangulated category as a set of topologizing(thick, Serre)subcategories satifying certain conditions.

But according to Ben-Zvi's answers in the question: Tannaka formalismhe proposed a definition of spectrum of a category as a pseudo-functor:i.e.

$SpecC: CRings\rightarrow FUNCT$ where $FUNCT$ is a 2-category of functors. He defined $SpecC(R)$ as category of functors from $C$ to $R-mod$.

What I am looking for is the reference talking about this stuff. According to the answers of Ben-Zvi. It seems that this POV to define spectrum has more chance to extend the reconstruction for schemes to reconstruction for stacks.

The functorial point of view of scheme is of course not due to Deligne. It is due to Gabriel

Thanks in advance!

All the related and unrelated comments are welcome

EDIT: Somebody point out the right reference should be Deligne's article in The Grothendieck Festschrift II. He mentioned the motivation is the categories of fiber functors form a gerbe on the fpqc site of the base scheme.(Deligne did not use **spectrum** there,though)