Refinement (category theory)

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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[edit]

Suppose is a category, an object in , and and two classes of morphisms in . The definition[1] of a refinement of in the class by means of the class consists of two steps.

Enrichment
  • A morphism in is called an enrichment of the object in the class of morphisms by means of the class of morphisms , if , and for any morphism from the class there exists a unique morphism in such that .
Refinement
  • An enrichment of the object in the class of morphisms by means of the class of morphisms is called a refinement of in by means of , if for any other enrichment (of in by means of ) there is a unique morphism in such that . The object is also called a refinement of in by means of .

Notations:

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

Similarly, if is a class of all morphisms whose ranges belong to a given class of objects in it is convenient to replace with in the notations (and in the terms):

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

Examples[edit]

  1. The bornologification[2][3] of a locally convex space is a refinement of in the category of locally convex spaces by means of the subcategory of normed spaces:
  2. The saturation[4][3] of a pseudocomplete[5] locally convex space is a refinement in the category of locally convex spaces by means of the subcategory of the Smith spaces:

See also[edit]

Notes[edit]

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

References[edit]

  • Kriegl, A.; Michor, P.W. (1997). The convenient setting of global analysis. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0780-3.