Quotient of an abelian category

In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category formula_1 by a Serre subcategory formula_2 is the abelian category formula_3 which, intuitively, is obtained from formula_1 by ignoring (i.e. treating as zero) all objects from formula_2. There is a canonical exact functor formula_6 whose kernel is formula_7.
Definition.
Formally, formula_8 is the category whose objects are those of formula_9 and whose morphisms from "X" to "Y" are given by the direct limit (of abelian groups) formula_10 over subobjects formula_11 and formula_12 such that formula_13 and formula_14. (Here, formula_15 and formula_16 denote quotient objects computed in formula_9.) Composition of morphisms in formula_3 is induced by the universal property of the direct limit. 
The canonical functor formula_19 sends an object "X" to itself and a morphism formula_20 to the corresponding element of the direct limit with "X′" = X and "Y′" = 0.
Examples.
Let formula_21 be a field and consider the abelian category formula_22 of all vector spaces over formula_21. Then the full subcategory formula_24 of finite-dimensional vector spaces is a Serre-subcategory of formula_22. The quotient formula_26 has as objects the formula_21-vector spaces, and the set of morphisms from formula_28 to formula_29 in formula_30 is formula_31 (which is a quotient of vector spaces). This has the effect of identifying all finite-dimensional vector spaces with 0, and of identifying two linear maps whenever their difference has finite-dimensional image.
Properties.
The quotient formula_8 is an abelian category, and the canonical functor formula_6 is exact. The kernel of formula_34 is formula_7, i.e., formula_36is a zero object of formula_3 if and only if formula_28 belongs to formula_2.
The quotient and canonical functor are characterized by the following universal property: if formula_40 is any abelian category and formula_41 is an exact functor such that formula_42 is a zero object of "formula_40" for each object formula_44, then there is a unique exact functor formula_45 such that formula_46.
Gabriel–Popescu.
The Gabriel–Popescu theorem states that any Grothendieck category formula_1 is equivalent to a quotient category formula_48, where formula_49denotes the abelian category of right modules over some unital ring formula_50, and formula_51 is some localizing subcategory of formula_49.