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 strong sense) iff with all objects contains all its subquotients and all extensions, i.e. 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 .
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