nLab topologizing subcategory

Definition

A full subcategory $T$ of an abelian category $A$ is topologizing if

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

• with any object, it contains all its subquotients in $A$

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 $A$. Given a (not necessarily unital) ring $R$, any reflective topologizing subcategory of $R$-$\mathrm{Mod}$ is coreflective.

Related concepts

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