nLab
neighborhood of a topologizing subcategory

Given a topologizing subcategory $𝕋$ of an abelian category $A$, the $n$-th neighborhood of $𝕋$ is the $n$-th power ${𝕋}^{(n)}$ of $𝕋$ with respect to the Gabriel multiplication of topologizing subcategories.

The union ${𝕋}^{(\mathrm{\infty })}:={\cup }_{n>1}{𝕋}^{(n)}$ is a topologizing subcategory of $A$ closed under extensions (that is a thick subcategory in the strong sense).

A typical example is when $𝕋={\Delta }_B$ is the minimal “subscheme” (= coreflective topologizing subcategory) of the category of additive endofunctors $A=\mathrm{End}B$ containing the identity functor. See differential monad.