nLab 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; called dense in Popescu) if it is closed under subobjects, quotients and extensions (in particular it is full and abelian). Some authors say Serre subcategory for a thick subcategory, though a stronger version of the notion of Serre subcategory may be appropriate (and is occasionally so defined) if the Abelian category is not the full subcategory of modules over a ring or ringoid (when the two notions agree).

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

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

where the colimit is over all $X',Y'$ in $A$ such that $Y'$ and $X/X'$ 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
• Chapter 15 in Carl Faith, Algebra: Rings, Modules, Categories, vol. 1, Grundlehren der mathematischen Wissenschaften 190, Springer (1973) [doi:10.1007/978-3-642-80634-6]
• N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp