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 $$K$$ is a category, $$X$$ an object in $$K$$, and $$\Gamma$$ and $$\Phi$$ two classes of morphisms in $$K$$. The definition of a refinement of $$X$$ in the class $$\Gamma$$ by means of the class $$\Phi$$ consists of two steps.

Notations:
 * A morphism $$\sigma:X'\to X$$ in $$K$$ is called an enrichment of the object $$X$$ in the class of morphisms $$\Gamma$$ by means of the class of morphisms $$\Phi$$, if $$\sigma\in\Gamma$$, and for any morphism $$\varphi:B\to X$$ from the class $$\Phi$$ there exists a unique morphism $$\varphi':B\to X'$$ in $$K$$ such that $$\varphi=\sigma\circ\varphi'$$.
 * An enrichment $$\rho:E\to X$$ of the object $$X$$ in the class of morphisms $$\Gamma$$ by means of the class of morphisms $$\Phi$$ is called a refinement of $$X$$ in $$\Gamma$$ by means of $$\Phi$$, if for any other enrichment $$\sigma:X'\to X$$ (of $$X$$ in $$\Gamma$$ by means of $$\Phi$$) there is a unique morphism $$\upsilon:E\to X'$$ in $$K$$ such that $$\rho=\sigma\circ\upsilon$$. The object $$E$$ is also called a refinement of $$X$$ in $$\Gamma$$ by means of $$\Phi$$.



\rho=\operatorname{ref}_\Phi^\Gamma X, \qquad E=\operatorname{Ref}_\Phi^\Gamma X. $$

In a special case when $$\Gamma$$ is a class of all morphisms whose ranges belong to a given class of objects $$L$$ in $$K$$ it is convenient to replace $$\Gamma$$ with $$L$$ in the notations (and in the terms):

\rho=\operatorname{ref}_\Phi^L X, \qquad E=\operatorname{Ref}_\Phi^L X. $$

Similarly, if $$\Phi$$ is a class of all morphisms whose ranges belong to a given class of objects $$M$$ in $$K$$ it is convenient to replace $$\Phi$$ with $$M$$ in the notations (and in the terms):

\rho=\operatorname{ref}_M^\Gamma X, \qquad E=\operatorname{Ref}_M^\Gamma X. $$

For example, one can speak about a refinement of $$X$$ in the class of objects $$L$$ by means of the class of objects $$M$$:



\rho=\operatorname{ref}_M^L X, \qquad E=\operatorname{Ref}_M^L X. $$

Examples

 * 1) The bornologification  $$X_{\operatorname{born}}$$ of a locally convex space $$X$$ is a refinement of $$X$$ in the category $$\operatorname{LCS}$$ of locally convex spaces by means of the subcategory $$\operatorname{Norm}$$ of normed spaces: $$X_{\operatorname{born}}=\operatorname{Ref}_{\operatorname{Norm}}^{\operatorname{LCS}}X$$
 * 2) The saturation $$X^\blacktriangle$$ of a pseudocomplete locally convex space $$X$$ is a refinement in the category $$\operatorname{LCS}$$ of locally convex spaces by means of the subcategory $$\operatorname{Smi}$$ of the Smith spaces: $$X^\blacktriangle=\operatorname{Ref}_{\operatorname{Smi}}^{\operatorname{LCS}}X$$