Quotient of an abelian category

In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category ${\displaystyle {\mathcal {A}}}$ by a Serre subcategory ${\displaystyle {\mathcal {B}}}$ is the abelian category ${\displaystyle {\mathcal {A}}/{\mathcal {B}}}$ which, intuitively, is obtained from ${\displaystyle {\mathcal {A}}}$ by ignoring (i.e. treating as zero) all objects from ${\displaystyle {\mathcal {B}}}$. There is a canonical exact functor ${\displaystyle Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal {B}}}$ whose kernel is ${\displaystyle {\mathcal {B}}}$, and ${\displaystyle {\mathcal {A}}/{\mathcal {B}}}$ is in a certain sense the most general abelian category with this property.

Forming Serre quotients of abelian categories is thus formally akin to forming quotients of groups. Serre quotients are somewhat similar to quotient categories, the difference being that with Serre quotients all involved categories are abelian and all functors are exact. Serre quotients also often have the character of localizations of categories, especially if the Serre subcategory is localizing.