# Quotient of an abelian 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 over subobjects and such that . The quotient *A/B* will then be an Abelian category, and there is a canonical functor sending an object *X* to itself and a morphism 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 is an exact functor such that *F(b)* is a zero object of *C* for each , then there is a unique exact functor such that .^{[1]}

## References

- ↑ Gabriel, Pierre,
*Des categories abeliennes*, Bull. Soc. Math. France**90**(1962), 323-448.

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