Nodal decomposition

Summary

In category theory, an abstract mathematical discipline, a nodal decomposition[1] of a morphism is a representation of as a product , where is a strong epimorphism,[2][3][4] a bimorphism, and a strong monomorphism.[5][3][4]

Nodal decomposition.

Uniqueness and notationsEdit

 
Uniqueness of the nodal decomposition.

If exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions   and   there exist isomorphisms   and   such that

 
 
 
 
Notations.

This property justifies some special notations for the elements of the nodal decomposition:

 

– here   and   are called the nodal coimage of  ,   and   the nodal image of  , and   the nodal reduced part of  .

In these notations the nodal decomposition takes the form

 

Connection with the basic decomposition in pre-abelian categoriesEdit

In a pre-abelian category   each morphism   has a standard decomposition

 ,

called the basic decomposition (here  ,  , and   are respectively the image, the coimage and the reduced part of the morphism  ).

 
Nodal and basic decompositions.

If a morphism   in a pre-abelian category   has a nodal decomposition, then there exist morphisms   and   which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:

 
 
 

Categories with nodal decompositionEdit

A category   is called a category with nodal decomposition[1] if each morphism   has a nodal decomposition in  . This property plays an important role in constructing envelopes and refinements in  .

In an abelian category   the basic decomposition

 

is always nodal. As a corollary, all abelian categories have nodal decomposition.

If a pre-abelian category   is linearly complete,[6] well-powered in strong monomorphisms[7] and co-well-powered in strong epimorphisms,[8] then   has nodal decomposition.[9]

More generally, suppose a category   is linearly complete,[6] well-powered in strong monomorphisms,[7] co-well-powered in strong epimorphisms,[8] and in addition strong epimorphisms discern monomorphisms[10] in  , and, dually, strong monomorphisms discern epimorphisms[11] in  , then   has nodal decomposition.[12]

The category Ste of stereotype spaces (being non-abelian) has nodal decomposition,[13] as well as the (non-additive) category SteAlg of stereotype algebras .[14]

NotesEdit

  1. ^ a b Akbarov 2016, p. 28.
  2. ^ An epimorphism   is said to be strong, if for any monomorphism   and for any morphisms   and   such that   there exists a morphism  , such that   and  .
     
  3. ^ a b Borceux 1994.
  4. ^ a b Tsalenko 1974.
  5. ^ A monomorphism   is said to be strong, if for any epimorphism   and for any morphisms   and   such that   there exists a morphism  , such that   and  
  6. ^ a b A category   is said to be linearly complete, if any functor from a linearly ordered set into   has direct and inverse limits.
  7. ^ a b A category   is said to be well-powered in strong monomorphisms, if for each object   the category   of all strong monomorphisms into   is skeletally small (i.e. has a skeleton which is a set).
  8. ^ a b A category   is said to be co-well-powered in strong epimorphisms, if for each object   the category   of all strong epimorphisms from   is skeletally small (i.e. has a skeleton which is a set).
  9. ^ Akbarov 2016, p. 37.
  10. ^ It is said that strong epimorphisms discern monomorphisms in a category  , if each morphism  , which is not a monomorphism, can be represented as a composition  , where   is a strong epimorphism which is not an isomorphism.
  11. ^ It is said that strong monomorphisms discern epimorphisms in a category  , if each morphism  , which is not an epimorphism, can be represented as a composition  , where   is a strong monomorphism which is not an isomorphism.
  12. ^ Akbarov 2016, p. 31.
  13. ^ Akbarov 2016, p. 142.
  14. ^ Akbarov 2016, p. 164.

ReferencesEdit

  • Borceux, F. (1994). Handbook of Categorical Algebra 1. Basic Category Theory. Cambridge University Press. ISBN 978-0521061193.
  • Tsalenko, M.S.; Shulgeifer, E.G. (1974). Foundations of category theory. Nauka.
  • Akbarov, S.S. (2016). "Envelopes and refinements in categories, with applications to functional analysis". Dissertationes Mathematicae. 513: 1–188. arXiv:1110.2013. doi:10.4064/dm702-12-2015. S2CID 118895911.