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

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

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

Created on June 8, 2011 at 16:01:50. See the history of this page for a list of all contributions to it.