Quotient of an abelian category


In mathematics, the quotient of an abelian category by a Serre subcategory is the abelian category which, intuitively, is obtained from by ignoring all objects from. There is a canonical exact functor whose kernel is.

Definition

Formally, is the category whose objects are those of and whose morphisms from X to Y are given by the direct limit over subobjects and such that and. Composition of morphisms in is induced by the universal property of the direct limit.
The canonical functor sends an object X to itself and a morphism to the corresponding element of the direct limit with X′ = X and Y′ = 0.

Examples

Let be a field and consider the abelian category of all vector spaces over. Then the full subcategory of finite-dimensional vector spaces is a Serre-subcategory of. The quotient has as objects the -vector spaces, and the set of morphisms from to in is . 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 is an abelian category, and the canonical functor is exact. The kernel of is, i.e., is a zero object of if and only if belongs to.
The quotient and canonical functor are characterized by the following universal property: if is any abelian category and is an exact functor such that is a zero object of for each object, then there is a unique exact functor such that.

Gabriel–Popescu

The Gabriel–Popescu theorem states that any Grothendieck category is equivalent to a quotient category, where denotes the abelian category of right modules over some unital ring, and is some localizing subcategory of.