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In
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. 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 An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...
.


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 In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
X is a refinement of X in the category \operatorname of locally convex spaces by means of the subcategory \operatorname of
normed space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
s: X_=\operatorname_^X # The saturation X^\blacktriangle of a pseudocompleteA
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
X is said to be ''pseudocomplete'' if each
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “si ...
Cauchy net In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize ...
in X converges.
locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
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 An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...


Notes


References

* * * {{Category theory Category theory Duality theories Functional analysis