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 $K$ is a category, $X$ an object in $K$, and $\Gamma$ and $\Phi$ two classes of morphisms in $K$. The definition of a refinement of $X$ in the class $\Gamma$ by means of the class $\Phi$ consists of two steps.

Enrichment

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

Refinement

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

Notations:

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

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

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

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

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

Examples

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

See also

• Envelope

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.