Refinement (category theory)


In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called envelope.

Definition

Suppose is a category, an object in, and and two classes of morphisms in. The definition of a refinement of in the class by means of the class consists of two steps.
  • A morphism in is called an enrichment of the object in the class of morphisms by means of the class of morphisms , if, and for any morphism from the class there exists a unique morphism in such that.
  • An enrichment of the object in the class of morphisms by means of the class of morphisms is called a refinement of in by means of , if for any other enrichment there is a unique morphism in such that. The object is also called a refinement of in by means of .
Notations:
In a special case when is a class of all morphisms whose ranges belong to a given class of objects in it is convenient to replace with in the notations :
Similarly, if is a class of all morphisms whose ranges belong to a given class of objects in it is convenient to replace with in the notations :
For example, one can speak about a refinement of in the class of objects by means of the class of objects :

Examples

  1. The bornologification of a locally convex space is a refinement of in the category of locally convex spaces by means of the subcategory of normed spaces:
  2. The saturation of a pseudocomplete locally convex space is a refinement in the category of locally convex spaces by means of the subcategory of the Smith spaces: