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 ${\displaystyle K}$ 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.

Enrichment

• 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 ${\displaystyle \varphi =\sigma \circ \varphi '}$.

Refinement

• 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

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 ${\displaystyle \operatorname {Norm} }$ of normed 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

• Envelope

Notes

1. Akbarov 2016, p. 52.
2. Kriegl & Michor 1997, p. 35.
3. Akbarov 2016, p. 57.
4. Akbarov 2003, p. 194.
5. A topological vector space ${\displaystyle X}$ is said to be pseudocomplete if each totally bounded Cauchy net in ${\displaystyle X}$ converges.

References

• 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.