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Refinement (category theory)

Suppose ${\displaystyle K}$ is a category, ${\displaystyle X}$ an object in ${\displaystyle K}$, and ${\displaystyle \Gamma }$ and ${\displaystyle \Phi }$ two classes of morphisms in ${\displaystyle K}$. The definition^[1] of a refinement of ${\displaystyle X}$ in the class ${\displaystyle \Gamma }$ by means of the class ${\displaystyle \Phi }$ consists of two steps.

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

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

Notations:

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

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

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

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

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

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

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

Examples[edit]

1. The bornologification^[2][3] ${\displaystyle X_{\operatorname {born} }}$ of a locally convex space ${\displaystyle X}$ is a refinement of ${\displaystyle X}$ in the category ${\displaystyle \operatorname {LCS} }$ of locally convex spaces by means of the subcategory ${\displaystyle \operatorname {Norm} }$ofnormed spaces: ${\displaystyle X_{\operatorname {born} }=\operatorname {Ref} _{\operatorname {Norm} }^{\operatorname {LCS} }X}$
2. The saturation^[4][3] ${\displaystyle X^{\blacktriangle }}$ of a pseudocomplete^[5] locally convex space ${\displaystyle X}$ is a refinement in the category ${\displaystyle \operatorname {LCS} }$ of locally convex spaces by means of the subcategory ${\displaystyle \operatorname {Smi} }$ of the Smith spaces: ${\displaystyle X^{\blacktriangle }=\operatorname {Ref} _{\operatorname {Smi} }^{\operatorname {LCS} }X}$

See also[edit]

• Envelope

Notes[edit]

^ Atopological vector space ${\displaystyle X}$ is said to be pseudocomplete if each totally bounded Cauchy netin${\displaystyle X}$ converges.

References[edit]

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

• 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. S2CID 115297067.

• Akbarov, S.S. (2016). "Envelopes and refinements in categories, with applications to functional analysis". Dissertationes Mathematicae. 513: 1–188. arXiv:1110.2013. doi:10.4064/dm702-12-2015. S2CID 118895911.