nLab
topologizing subcategory

Contents

Definition

A full subcategory TT of an abelian category AA is topologizing if

  • it is closed with respect to finite coproducts (taken in AA)

  • with any object, it contains all its subquotients in AA

In particular, it is nonempty: it contains a zero object (which equals to the coproduct of the empty set of objects).

A topologizing subcategory is a thick subcategory in strong sense if it is also closed under extensions.

The terminology topologizing subcategory is (probably) coming from the related notion of a topologizing filter from the localization theory of rings.

Properties

The classes of topologizing subcategories, reflective topologizing subcategories and coreflective topologizing subcategories are closed under Gabriel multiplication defined on the class of full subcategories of AA. Given a (not necessarily unital) ring RR, any reflective topologizing subcategory of RR-Mod\mathrm{Mod} is coreflective.

Related entries include defining ideal of a topologizing subcategory, neighborhood of a topologizing subcategory, Gabriel multiplication, thick subcategory, Serre subcategory, local abelian category, differential monad, localization of an abelian category.

References

  • 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

  • V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf

Revised on November 19, 2011 15:18:52 by Urs Schreiber (82.113.99.53)