neighborhood of a topologizing subcategory

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

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

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

Last revised on May 5, 2011 at 15:44:34. See the history of this page for a list of all contributions to it.