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In [[Category theory|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 (category theory)|envelope]]. 
== Definition ==

Suppose <math>K</math> is a category, <math>X</math> an object in <math>K</math>, and <math>\Gamma</math> and <math>\Phi</math> two classes of morphisms in <math>K</math>. The definition{{sfn|Akbarov|2016|p=52}} of a refinement of <math>X</math> in the class <math>\Gamma</math> by means of the class <math>\Phi</math> consists of two steps.
[[File:Enrichment-2.jpg|thumb|Enrichment]]

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

: <math>
\rho=\operatorname{ref}_\Phi^\Gamma X, \qquad 
E=\operatorname{Ref}_\Phi^\Gamma X.
</math>

In a special case when <math>\Gamma</math> is a class of all morphisms whose ranges belong to a given class of objects <math>L</math> in <math>K</math> it is convenient to replace <math>\Gamma</math> with <math>L</math> in the notations (and in the terms):
 
: <math>
\rho=\operatorname{ref}_\Phi^L X, \qquad 
E=\operatorname{Ref}_\Phi^L X.
</math>

Similarly, if <math>\Phi</math> is a class of all morphisms whose ranges belong to a given class of objects <math>M</math> in <math>K</math> it is convenient to replace <math>\Phi</math> with <math>M</math> in the notations (and in the terms):
 
: <math>
\rho=\operatorname{ref}_M^\Gamma X, \qquad 
E=\operatorname{Ref}_M^\Gamma X.
</math>

For example, one can speak about a ''refinement of <math>X</math> in the class of objects <math>L</math> by means of the class of objects <math>M</math>'':

: <math>
\rho=\operatorname{ref}_M^L X, \qquad 
E=\operatorname{Ref}_M^L X.
</math>

== Examples ==

# The '''bornologification'''{{sfn|Kriegl|Michor|1997|p=35}}{{sfn|Akbarov|2016|p=57}}  <math>X_{\operatorname{born}}</math> of a locally convex space <math>X</math> is a refinement of <math>X</math> in the category <math>\operatorname{LCS}</math> of locally convex spaces by means of the subcategory <math>\operatorname{Norm}</math> of normed spaces: <math>X_{\operatorname{born}}=\operatorname{Ref}_{\operatorname{Norm}}^{\operatorname{LCS}}X</math>
# The '''saturation'''{{sfn|Akbarov|2003|p=194}}{{sfn|Akbarov|2016|p=57}} <math>X^\blacktriangle</math> of a pseudocomplete<ref>A [[Topological vector space|topological vector space]] <math>X</math> is said to be ''pseudocomplete'' if each totally bounded [[Net (mathematics)#Cauchy nets|Cauchy net]] in <math>X</math> converges.</ref> locally convex space <math>X</math> is a refinement in the category <math>\operatorname{LCS}</math> of locally convex spaces by means of the subcategory <math>\operatorname{Smi}</math> of the [[Smith space]]s: <math>X^\blacktriangle=\operatorname{Ref}_{\operatorname{Smi}}^{\operatorname{LCS}}X</math>

==See also==
*[[Envelope (category theory)|Envelope]]

==Notes==
{{reflist}}

== References ==
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* {{cite book |last1=Kriegl |first1=A. | last2=Michor |first2=P.W. |date= 1997 |title= The convenient setting of global analysis |url= https://bookstore.ams.org/surv-53 |location= Providence, Rhode Island |publisher= American Mathematical Society |isbn=0-8218-0780-3}}

*{{cite journal|last=Akbarov|first=S.S.|title=Pontryagin duality in the theory of topological vector spaces and in topological algebra|journal=Journal of Mathematical Sciences|year=2003|volume=113|issue=2|pages=179–349|doi=10.1023/A:1020929201133|s2cid=115297067|doi-access=free}}

*{{cite journal|last=Akbarov|first=S.S.|title=Envelopes and refinements in categories, with applications to functional analysis|url=https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513|journal=Dissertationes Mathematicae|year=2016|volume=513|pages=1–188|arxiv=1110.2013|doi=10.4064/dm702-12-2015|s2cid=118895911}}

{{Functional Analysis}}
{{Category theory}}

[[Category:Category theory]]
[[Category:Duality theories]]
[[Category:Functional analysis]]

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