Refinement (category theory)

Suppose $K$ is a category, $X$ an object in $K$, and $\backslash Gamma$ and $\backslash Phi$ two classes of morphisms in $K$. The definition{{sfn|Akbarov|2016|p=52}} of a refinement of $X$ in the class $\backslash Gamma$ by means of the class $\backslash Phi$ consists of two steps.

File:Enrichment-2.jpg

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

File:Refinement.jpg

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

Notations:

: $$

\rho=\operatorname{ref}_\Phi^\Gamma X, \qquad

E=\operatorname{Ref}_\Phi^\Gamma X.

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

: $$

\rho=\operatorname{ref}_\Phi^L X, \qquad

E=\operatorname{Ref}_\Phi^L X.

Similarly, if $\backslash Phi$ is a class of all morphisms whose ranges belong to a given class of objects $M$ in $K$ it is convenient to replace $\backslash Phi$ 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.

1. The bornologification{{sfn|Kriegl|Michor|1997|p=35}}{{sfn|Akbarov|2016|p=57}} $X\_\{\backslash operatorname\{born\}\}$ of a locally convex space $X$ is a refinement of $X$ in the category $\backslash operatorname\{LCS\}$ of locally convex spaces by means of the subcategory $\backslash operatorname\{Norm\}$ of normed spaces: $X\_\{\backslash operatorname\{born\}\}=\backslash operatorname\{Ref\}\_\{\backslash operatorname\{Norm\}\}^\{\backslash operatorname\{LCS\}\}X$
2. The saturation{{sfn|Akbarov|2003|p=194}}{{sfn|Akbarov|2016|p=57}} $X^\backslash blacktriangle$ of a pseudocompleteA topological vector space $X$ is said to be pseudocomplete if each totally bounded Cauchy net in $X$ converges. locally convex space $X$ is a refinement in the category $\backslash operatorname\{LCS\}$ of locally convex spaces by means of the subcategory $\backslash operatorname\{Smi\}$ of the Smith spaces: $X^\backslash blacktriangle=\backslash operatorname\{Ref\}\_\{\backslash operatorname\{Smi\}\}^\{\backslash operatorname\{LCS\}\}X$

• Envelope

{{reflist}}

• {{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