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

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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

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  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

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Notes

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  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

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