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 formula_1 is a category, formula_2 an object in formula_1, and formula_4 and formula_5 two classes of morphisms in formula_1. The definition of a refinement of formula_2 in the class formula_4 by means of the class formula_5 consists of two steps.
Notations:
In a special case when formula_4 is a class of all morphisms whose ranges belong to a given class of objects formula_41 in formula_1 it is convenient to replace formula_4 with formula_41 in the notations (and in the terms):
Similarly, if formula_5 is a class of all morphisms whose ranges belong to a given class of objects formula_47 in formula_1 it is convenient to replace formula_5 with formula_47 in the notations (and in the terms):
For example, one can speak about a "refinement of formula_2 in the class of objects formula_41 by means of the class of objects formula_47":