# nLab topologizing subcategory

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## Applications

#### Notions of subcategory

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# Contents

## 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.

## 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

Last revised on November 19, 2011 at 15:18:52. See the history of this page for a list of all contributions to it.