Sometimes the same definition is used in abelian categories as well. However, for many authors, including Pierre Gabriel, in abelian categories, this term denotes the stronger notion of a topologizing subcategory closed under extensions; in other words, a nonempty full subcategory of an abelian category is thick (in the strong sense) iff for every exact sequence
in , the object is in iff and are in .
For some authors the thick subcategory (strong version) is called a Serre subcategory (in a weak sense), the term which we reserve for (generally) a stronger notion.
For any subcategory of an abelian category one denotes by the full subcategory of generated by all objects for which any (nonzero) subquotient of in has a (nonzero) subobject from . This becomes an idempotent operation on the class of subcategories of where iff is topologizing. Moreover is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation .
Following the extensions of an early work of Serre by Grothendieck and Gabriel, for a thick subcategory in an abelian category , one defines the (Serre) quotient category has the same objects as and
where the colimit runs through all subobjects , such that , . The quotient functor is obvious.
Notice that the set of morphisms is small, so that the Serre quotient category exists. On the other hand, one can construct an equivalent localization by the Gabriel-Zisman localizing at the class of all morphisms whose kernel and cokernel are in . Although admits the calculus of fractions, this method does not guarantee the existence in general.
A thick subcategory (here always in strong sense) is said to be localizing if and admits a right adjoint , often called the section functor. In other words is a reflective subcategory of . Every coreflective thick subcategory admits a section functor, and the converse holds if has injective envelopes. A thick subcategory is a coreflective iff is a torsion theory where
A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9
Springer eom: localization of categories