nLab neighborhood of a topologizing subcategory

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

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

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