Quotient of an abelian category

Formally, ${\displaystyle {\mathcal {A}}/{\mathcal {B}}}$ is the category whose objects are those of ${\displaystyle {\mathcal {A}}}$ and whose morphisms from X to Y are given by the direct limit (of abelian groups) ${\displaystyle \varinjlim \mathrm {Hom} _{\mathcal {A}}(X',Y/Y')}$ over subobjects ${\displaystyle X'\subseteq X}$ and ${\displaystyle Y'\subseteq Y}$ such that ${\displaystyle X/X'\in {\cal {B}}}$ and ${\displaystyle Y'\in {\cal {B}}}$. (Here, ${\displaystyle X/X'}$ and ${\displaystyle Y/Y'}$ denote quotient objects computed in ${\displaystyle {\mathcal {A}}}$.) Composition of morphisms in ${\displaystyle {\mathcal {A}}/{\mathcal {B}}}$ is induced by the universal property of the direct limit.

The canonical functor ${\displaystyle Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal {B}}}$ sends an object X to itself and a morphism ${\displaystyle f\colon X\to Y}$ to the corresponding element of the direct limit with X′ = X and Y′ = 0.

Let ${\displaystyle k}$ be a field and consider the abelian category ${\displaystyle {\rm {Mod}}(k)}$ of all vector spaces over ${\displaystyle k}$. Then the full subcategory ${\displaystyle {\rm {mod}}(k)}$ of finite-dimensional vector spaces is a Serre-subcategory of ${\displaystyle {\rm {Mod}}(k)}$. The quotient ${\displaystyle {\cal {{C}={\rm {Mod}}(k)/{\rm {mod}}(k)}}}$ has as objects the ${\displaystyle k}$-vector spaces, and the set of morphisms from ${\displaystyle X}$ to ${\displaystyle Y}$ in ${\displaystyle {\cal {C}}}$ is

$${\displaystyle \{k{\text{-linear maps from }}X{\text{ to }}Y\}/\{k{\text{-linear maps from }}X{\text{ to }}Y{\text{ with finite-dimensional image}}\}}$$

(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.

The quotient ${\displaystyle {\mathcal {A}}/{\mathcal {B}}}$ is an abelian category, and the canonical functor ${\displaystyle Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal {B}}}$ is exact. The kernel of ${\displaystyle Q}$ is ${\displaystyle {\mathcal {B}}}$, i.e., ${\displaystyle Q(X)}$is a zero object of ${\displaystyle {\mathcal {A}}/{\mathcal {B}}}$ if and only if ${\displaystyle X}$ belongs to ${\displaystyle {\mathcal {B}}}$.

The quotient and canonical functor are characterized by the following universal property: if ${\displaystyle {\mathcal {C}}}$ is any abelian category and ${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {C}}}$ is an exact functor such that ${\displaystyle F(X)}$ is a zero object of ${\displaystyle {\mathcal {C}}}$ for each object ${\displaystyle X\in {\mathcal {B}}}$, then there is a unique exact functor ${\displaystyle {\overline {F}}\colon {\mathcal {A}}/{\mathcal {B}}\to {\mathcal {C}}}$ such that ${\displaystyle F={\overline {F}}\circ Q}$.[1]

The Gabriel–Popescu theorem states that any Grothendieck category ${\displaystyle {\mathcal {A}}}$ is equivalent to a quotient category ${\displaystyle \operatorname {Mod} (R)/{\cal {B}}}$, where ${\displaystyle \operatorname {Mod} (R)}$denotes the abelian category of right modules over some unital ring ${\displaystyle R}$, and ${\displaystyle {\cal {B}}}$ is some localizing subcategory of ${\displaystyle \operatorname {Mod} (R)}$.[2]