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 $\mathrm{\Gamma }$ and $\mathrm{\Phi }$ two classes of morphisms in $K$. The definition of a refinement of $X$ in the class $\mathrm{\Gamma }$ by means of the class $\mathrm{\Phi }$ consists of two steps.

Enrichment

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

Refinement

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

Notations:

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

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

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

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

$\rho ={\mathrm{ref}}_M^{\mathrm{\Gamma }}X,E={\mathrm{Ref}}_M^{\mathrm{\Gamma }}X.$

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

$\rho ={\mathrm{ref}}_M^{L}X,E={\mathrm{Ref}}_M^{L}X.$

Examples

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

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. https://bookstore.ams.org/surv-53. 

• 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. 

• Akbarov, S.S. (2016). "Envelopes and refinements in categories, with applications to functional analysis". Dissertationes Mathematicae 513: 1–188. doi:10.4064/dm702-12-2015. https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513.