Serre quotient category

In mathematics, the quotient of an abelian category A by a Serre subcategory B is the category whose objects are those of A and whose morphisms from X to Y are given by the direct limit ${\displaystyle \varinjlim \mathrm {Hom} _{A}(X',Y/Y')}$ over subobjects ${\displaystyle X'\subseteq X}$ and ${\displaystyle Y'\subseteq Y}$ such that ${\displaystyle X/X',Y'\in B}$. The quotient A/B will then be an Abelian category, and there is a canonical functor ${\displaystyle Q\colon A\to A/B}$ sending 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. This Abelian quotient satisfies the universal property that if C is any other Abelian category, and ${\displaystyle F\colon A\to C}$ is an exact functor such that F(b) is a zero object of C for each ${\displaystyle b\in B}$, then there is a unique exact functor ${\displaystyle {\overline {F}}\colon A/B\to C}$ such that ${\displaystyle F={\overline {F}}\circ Q}$.

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