Interesting questions.
Actually, this is indeed related to work on defining a natural topology on categories, which is part of noncommutative algebraic geometry.

A. Rosenberg defined the *left spectrum* for a noncommutative ring in 1981 (see The left spectrum, the Levitzki radical, and noncommutative schemes), and further generalized this spectrum to any abelian category (see reconstruction of schemes), and proved the so called Gabriel-Rosenberg reconstruction theorem which led to the correct definition of noncommutative scheme. I might have time to talk about this later. But for now, I shall just point out some papers, such as Spectra of noncommutative spaces.

In this paper, Rosenberg takes an abelian category as a "noncommutative space" and defines various spectra for different goals. (**ONE remarkable destination is for representation theory of Lie algebras and quantum groups.**)

One can not only define spectrum for abelian categories; **this notion also makes sense in a non-abelian category and a triangulated category.** In the paper Spectra related with localizations, Rosenberg defined the spectrum directly related to localization of categories. Roughly speaking, the spectrum of a category is a family of topologizing subcategories (which by definition, are closed under direct sum, sub- and quotient; in particular, thick or Serre subcategories) satifying some additional conditions.

There is also another paper, Underlying spaces of noncommutative schemes, trying to investigate the underlying space of a noncommutative scheme or other noncommutative "space" in noncommutative algebraic geometry. If we want to save flat descent in general, we might lose the base change property. In this work, Rosenberg deals with the "quasi-topology" (which means dropping the base change property) and defines the *associative spectrum* of a category.
Moreover: for the goals of representation theory, he built a framework **relating representation theory with the spectrum of abelian category** (in particular, categories of modules). Actually, in this language, irreducible representations are in one-to-one correspondence with the closed points in the spectrum; generic points in the spectrum also produce representations (not necessarily irreducible).

The most important part in this work is that it provided a completely categorical (algebro-geometric) way to do induction in an abelian category instead of the derived category. (I will explain this later if I have time). This semester, Rosenberg gave us a lecture course, **using this framework to compute all the irreducible representations** for the Weyl algebra, the enveloping algebra, quantized enveloping algebras, algebras of differential operators, $SL\_2({\mathbb R})$ and other algebraic groups, or related associative algebras. It works very efficiently. For example, computing irreducible representations of $U(sl_3)$ is believed to be very complicated, but using this spectrum framework, it becomes much simpler.

The general framework for these is contained in the paper Spectra, associated points and representation theory. If you want to see some concrete examples using this machine, you should look at Rosenberg's old book *Noncommutative Algebraic Geometry And Representations Of Quantized Algebras.*
There is another paper Spectra of `spaces' represented by abelian categories, providing the general theory for this machinery.

Furthermore, we can define the spectrum for an *exact category*; even more generally, for any Grothendieck site, and so for any category (because any category has a canonical Grothendieck pretopology). Rosenberg has recent work defining the spectrum for such categories -- Geometry of right exact `spaces' -- the main motivation for this work is to provide a background for higher universal algebraic K-theory for a *right exact category* (a category with a family of strict epimorphisms can be taken as a one-sided exact category). **More important motivation is to study algebraic cycles for noncommutative schemes.** (Warning: this paper is very abstract and hard to read. We will go through this paper in the lecture course this semester.)

All of these things will appear soon in his new book with Konstevich (but I am not sure of the exact time). If I have enough time to post, I will explain in more detail, how the theory of the spectrum for abelian categories comes into representation theory, and how this picture is related to the derived picture of Beilinson-Bernstein and Deligne.
In fact, today we have just learned Beck's theorem for Karoubian triangulated categories and will do the DG-version of Beck's theorem later. And then he will introduce the spectrum for triangulated categories, and explain the noncommutative algebraic geometry facts behind the BBD machine and the connection with his abelian machine.

14more comments