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.

* 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 $\varphi=\sigma\circ\varphi'$.

* 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:

: $\rho=\operatorname_\Phi^\Gamma X, \qquad E=\operatorname_\Phi^\Gamma X.$

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):

: $\rho=\operatorname_\Phi^L X, \qquad E=\operatorname_\Phi^L X.$

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):

: $\rho=\operatorname_M^\Gamma X, \qquad E=\operatorname_M^\Gamma X.$

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

: $\rho=\operatorname_M^L X, \qquad E=\operatorname_M^L X.$

Examples

# The bornologification $X_$ of a locally convex space $X$ is a refinement of $X$ in the category $\operatorname$ of locally convex spaces by means of the subcategory $\operatorname$ of normed spaces: $X_=\operatorname_^X$
# The saturation $X^\blacktriangle$ 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 $\operatorname$ of locally convex spaces by means of the subcategory $\operatorname$ of the Smith spaces: $X^\blacktriangle=\operatorname_^X$

See also

*Envelope

Notes

References

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{{Category theory

Category theory
Duality theories
Functional analysis