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)

  • $\begingroup$ I misinterpreted the question, so I've deleted my answer. Shizhuo, would you mind clarifying what you mean a little bit? $\endgroup$ – Harry Gindi Mar 2 '10 at 7:40
  • $\begingroup$ Ok,see the answers given by Ben-Zvi. He defined spectrum of a category C as a stack,say $SpecC:CRings\rightarrow FUNCT$ where FUNCT is 2-category of functors:$C\rightarrow R-mod$(sloppy notions) $\endgroup$ – Shizhuo Zhang Mar 2 '10 at 8:04
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    $\begingroup$ Viewing a scheme as a functor was definitely not first done by Deligne. It's all over the place in EGA (though not as much as it should be!), and probably even in Grothendieck's seminar Bourbaki talks. It might even go back to Cartier in the brief pre-Grothendieck stage of scheme theory. $\endgroup$ – JBorger Mar 2 '10 at 20:59
  • $\begingroup$ No, what I am asking is not functorial view of point of scheme..... $\endgroup$ – Shizhuo Zhang Mar 2 '10 at 22:09
  • $\begingroup$ Oops. Sorry. I need to start reading these more carefully... $\endgroup$ – JBorger Mar 3 '10 at 2:24

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