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 \varinjlim \mathrm{Hom}_A(X', Y/Y') over subobjects X' \subseteq X and Y' \subseteq Y such that X/X', Y' \in B. The quotient A/B will then be an Abelian category, and there is a canonical functor Q \colon A \to A/B sending an object X to itself and a morphism 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 F \colon A \to C is an exact functor such that F(b) is a zero object of C for each b \in B, then there is a unique exact functor \overline{F} \colon A/B \to C such that F = \overline{F} \circ Q.[1]

References

  1. Gabriel, Pierre, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448.
This article is issued from Wikipedia - version of the Thursday, December 24, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.