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]}

## Uniqueness and notations

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

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 categories

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 ).

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 decomposition

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]}

## Notes

- ^
^{a}^{b}Akbarov 2016, p. 28. **^**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 .- ^
^{a}^{b}Borceux 1994. - ^
^{a}^{b}Tsalenko 1974. **^**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- ^
^{a}^{b}A category is said to be*linearly complete*, if any functor from a linearly ordered set into has direct and inverse limits. - ^
^{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). - ^
^{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). **^**Akbarov 2016, p. 37.**^**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.**^**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.**^**Akbarov 2016, p. 31.**^**Akbarov 2016, p. 142.**^**Akbarov 2016, p. 164.

## References

- 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.