# Applications and examples of quotient categories of abelian categories

I want to get motivated to learn more about quotient categories of abelian categories by a Serre subcategory or even by a localizing category as they are described in Pierre Gabriel's thesis "Des catégories abéliennes".

So I am looking for some applications of this theory or theorems, which can be proven nicely by using quotient categories.

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I believe that Serre's "mod $\mathcal{C}$" theory is a nice application of these ideas in topology. –  Akhil Mathew May 11 at 17:58

Recent work of Nick Katz (see his book "Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms", available on www.math.princeton.edu/~nmk) depends quite crucially on the properties of a quotient abelian category (of perverse sheaves on the multiplicative group) defined and studied previously by Gabber and Loeser; the quotient has the effect of allowing a certain convolution operation to be used to define a Tannakian structure, as Katz explains in Chapter 2 of his book.

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Have you considered registering one of your five accounts? –  S. Carnahan May 12 at 12:14
Serre's classes also have applications in profinite group theory (pro-$\mathcal{C}$-groups), but that uses the localisation less directly.