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Refinement (category theory)

释义

Definition
Examples
See also
Notes
References

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{{sfn|Akbarov|2016|p=52}} 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 (of in by means of ) 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 (and in the terms):

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 (and in the terms):

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

Examples

The bornologification{{sfn|Kriegl|Michor|1997|p=35}}{{sfn|Akbarov|2016|p=57}} of a locally convex space is a refinement of in the category of locally convex spaces by means of the subcategory of normed spaces:
The saturation{{sfn|Akbarov|2003|p=194}}{{sfn|Akbarov|2016|p=57}} of a pseudocomplete^&91;1&93; locally convex space is a refinement in the category of locally convex spaces by means of the subcategory of the Smith spaces:

Notes

1. ^A topological vector space

is said to be pseudocomplete if each totally bounded Cauchy net in

converges.

References

{{cite book |last=Kriegl |first=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| ref = harv}}

{{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|url=http://www.springerlink.com/content/k62m72960101g6q2/| ref = harv}}

{{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| ref = harv}}

3 : Category theory|Duality theories|Functional analysis

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Definition

Examples

See also

Notes

References