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 [math]\displaystyle{ K }[/math] is a category, [math]\displaystyle{ X }[/math] an object in [math]\displaystyle{ K }[/math], and [math]\displaystyle{ \Gamma }[/math] and [math]\displaystyle{ \Phi }[/math] two classes of morphisms in [math]\displaystyle{ K }[/math]. The definition of a refinement of [math]\displaystyle{ X }[/math] in the class [math]\displaystyle{ \Gamma }[/math] by means of the class [math]\displaystyle{ \Phi }[/math] consists of two steps.

Enrichment

• A morphism [math]\displaystyle{ \sigma:X'\to X }[/math] in [math]\displaystyle{ K }[/math] is called an enrichment of the object [math]\displaystyle{ X }[/math] in the class of morphisms [math]\displaystyle{ \Gamma }[/math] by means of the class of morphisms [math]\displaystyle{ \Phi }[/math], if [math]\displaystyle{ \sigma\in\Gamma }[/math], and for any morphism [math]\displaystyle{ \varphi:B\to X }[/math] from the class [math]\displaystyle{ \Phi }[/math] there exists a unique morphism [math]\displaystyle{ \varphi':B\to X' }[/math] in [math]\displaystyle{ K }[/math] such that [math]\displaystyle{ \varphi=\sigma\circ\varphi' }[/math].

Refinement

• An enrichment [math]\displaystyle{ \rho:E\to X }[/math] of the object [math]\displaystyle{ X }[/math] in the class of morphisms [math]\displaystyle{ \Gamma }[/math] by means of the class of morphisms [math]\displaystyle{ \Phi }[/math] is called a refinement of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ \Gamma }[/math] by means of [math]\displaystyle{ \Phi }[/math], if for any other enrichment [math]\displaystyle{ \sigma:X'\to X }[/math] (of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ \Gamma }[/math] by means of [math]\displaystyle{ \Phi }[/math]) there is a unique morphism [math]\displaystyle{ \upsilon:E\to X' }[/math] in [math]\displaystyle{ K }[/math] such that [math]\displaystyle{ \rho=\sigma\circ\upsilon }[/math]. The object [math]\displaystyle{ E }[/math] is also called a refinement of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ \Gamma }[/math] by means of [math]\displaystyle{ \Phi }[/math].

Notations:

[math]\displaystyle{ \rho=\operatorname{ref}_\Phi^\Gamma X, \qquad E=\operatorname{Ref}_\Phi^\Gamma X. }[/math]

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

[math]\displaystyle{ \rho=\operatorname{ref}_\Phi^L X, \qquad E=\operatorname{Ref}_\Phi^L X. }[/math]

Similarly, if [math]\displaystyle{ \Phi }[/math] is a class of all morphisms whose ranges belong to a given class of objects [math]\displaystyle{ M }[/math] in [math]\displaystyle{ K }[/math] it is convenient to replace [math]\displaystyle{ \Phi }[/math] with [math]\displaystyle{ M }[/math] in the notations (and in the terms):

[math]\displaystyle{ \rho=\operatorname{ref}_M^\Gamma X, \qquad E=\operatorname{Ref}_M^\Gamma X. }[/math]

For example, one can speak about a refinement of [math]\displaystyle{ X }[/math] in the class of objects [math]\displaystyle{ L }[/math] by means of the class of objects [math]\displaystyle{ M }[/math]:

[math]\displaystyle{ \rho=\operatorname{ref}_M^L X, \qquad E=\operatorname{Ref}_M^L X. }[/math]

Examples

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

See also

• Envelope

References

• Kriegl, A.; Michor, P.W. (1997). The convenient setting of global analysis. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0780-3. https://bookstore.ams.org/surv-53. 

• Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences 113 (2): 179–349. doi:10.1023/A:1020929201133. 

• Akbarov, S.S. (2016). "Envelopes and refinements in categories, with applications to functional analysis". Dissertationes Mathematicae 513: 1–188. doi:10.4064/dm702-12-2015. https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513.