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localization of an abelian category

Abstract localization functors among abelian categories have several descriptions. Additional descriptions exist if in addition the category is Grothendieck.

A nonempty subcategory of an abelian category is thick (in the sense of Pierre Gabriel) if it is closed under subobjects, quotients and extensions. In the special case of the abelian categories of $R$-modules (where $R$ is a ring) this agrees with the notion of a Serre subcategory (in general the latter is a stronger notion).

Following Jean-Pierre Serre, given a thick subcategory $T$, define the quotient category $A/T$ whose objects are the objects of $A$ and where the morphisms in $A/T$ are defined by

$${\mathrm{Hom}}_{A/T}(X,Y):=\mathrm{colim}{\mathrm{Hom}}_A(X\prime ,Y/Y\prime ),$$\mathrm{Hom}_{A/T}(X,Y) := \mathrm{colim}\, \mathrm{Hom}_A(X',Y/Y'),

where the colimit is over all $X\prime ,Y\prime$ in $A$ such that $Y\prime$ and $X/X\prime$ are in $T$. There is a canonical quotient functor $Q:A\to A/T$ which is the identity on objects. The quotient category $A/T$ is abelian.

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448, numdam
• Francis Borceux, Handbook of categorical algebra