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Nodal decomposition

## Summary

In category theory, an abstract mathematical discipline, a nodal decomposition[1] of a morphism ${\displaystyle \varphi :X\to Y}$ is a representation of ${\displaystyle \varphi }$ as a product ${\displaystyle \varphi =\sigma \circ \beta \circ \pi }$, where ${\displaystyle \pi }$ is a strong epimorphism,[2][3][4] ${\displaystyle \beta }$ a bimorphism, and ${\displaystyle \sigma }$ a strong monomorphism.[5][3][4]

Nodal decomposition.

## Uniqueness and notations

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 ${\displaystyle \varphi =\sigma \circ \beta \circ \pi }$  and ${\displaystyle \varphi =\sigma '\circ \beta '\circ \pi '}$  there exist isomorphisms ${\displaystyle \eta }$  and ${\displaystyle \theta }$  such that

${\displaystyle \pi '=\eta \circ \pi ,}$
${\displaystyle \beta =\theta \circ \beta '\circ \eta ,}$
${\displaystyle \sigma '=\sigma \circ \theta .}$

Notations.

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

{\displaystyle {\begin{aligned}&\pi =\operatorname {coim} _{\infty }\varphi ,&&P=\operatorname {Coim} _{\infty }\varphi ,\\&\beta =\operatorname {red} _{\infty }\varphi ,&&\\&\sigma =\operatorname {im} _{\infty }\varphi ,&&Q=\operatorname {Im} _{\infty }\varphi ,\end{aligned}}}

– here ${\displaystyle \operatorname {coim} _{\infty }\varphi }$  and ${\displaystyle \operatorname {Coim} _{\infty }\varphi }$  are called the nodal coimage of ${\displaystyle \varphi }$ , ${\displaystyle \operatorname {im} _{\infty }\varphi }$  and ${\displaystyle \operatorname {Im} _{\infty }\varphi }$  the nodal image of ${\displaystyle \varphi }$ , and ${\displaystyle \operatorname {red} _{\infty }\varphi }$  the nodal reduced part of ${\displaystyle \varphi }$ .

In these notations the nodal decomposition takes the form

${\displaystyle \varphi =\operatorname {im} _{\infty }\varphi \circ \operatorname {red} _{\infty }\varphi \circ \operatorname {coim} _{\infty }\varphi .}$

## Connection with the basic decomposition in pre-abelian categories

In a pre-abelian category ${\displaystyle {\mathcal {K}}}$  each morphism ${\displaystyle \varphi }$  has a standard decomposition

${\displaystyle \varphi =\operatorname {im} \varphi \circ \operatorname {red} \varphi \circ \operatorname {coim} \varphi }$ ,

called the basic decomposition (here ${\displaystyle \operatorname {im} \varphi =\ker(\operatorname {coker} \varphi )}$ , ${\displaystyle \operatorname {coim} \varphi =\operatorname {coker} (\ker \varphi )}$ , and ${\displaystyle \operatorname {red} \varphi }$  are respectively the image, the coimage and the reduced part of the morphism ${\displaystyle \varphi }$ ).

Nodal and basic decompositions.

If a morphism ${\displaystyle \varphi }$  in a pre-abelian category ${\displaystyle {\mathcal {K}}}$  has a nodal decomposition, then there exist morphisms ${\displaystyle \eta }$  and ${\displaystyle \theta }$  which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:

${\displaystyle \operatorname {coim} _{\infty }\varphi =\eta \circ \operatorname {coim} \varphi ,}$
${\displaystyle \operatorname {red} \varphi =\theta \circ \operatorname {red} _{\infty }\varphi \circ \eta ,}$
${\displaystyle \operatorname {im} _{\infty }\varphi =\operatorname {im} \varphi \circ \theta .}$

## Categories with nodal decomposition

A category ${\displaystyle {\mathcal {K}}}$  is called a category with nodal decomposition[1] if each morphism ${\displaystyle \varphi }$  has a nodal decomposition in ${\displaystyle {\mathcal {K}}}$ . This property plays an important role in constructing envelopes and refinements in ${\displaystyle {\mathcal {K}}}$ .

In an abelian category ${\displaystyle {\mathcal {K}}}$  the basic decomposition

${\displaystyle \varphi =\operatorname {im} \varphi \circ \operatorname {red} \varphi \circ \operatorname {coim} \varphi }$

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

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

More generally, suppose a category ${\displaystyle {\mathcal {K}}}$  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 ${\displaystyle {\mathcal {K}}}$ , and, dually, strong monomorphisms discern epimorphisms[11] in ${\displaystyle {\mathcal {K}}}$ , then ${\displaystyle {\mathcal {K}}}$  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

1. ^ a b Akbarov 2016, p. 28.
2. ^ An epimorphism ${\displaystyle \varepsilon :A\to B}$  is said to be strong, if for any monomorphism ${\displaystyle \mu :C\to D}$  and for any morphisms ${\displaystyle \alpha :A\to C}$  and ${\displaystyle \beta :B\to D}$  such that ${\displaystyle \beta \circ \varepsilon =\mu \circ \alpha }$  there exists a morphism ${\displaystyle \delta :B\to C}$ , such that ${\displaystyle \delta \circ \varepsilon =\alpha }$  and ${\displaystyle \mu \circ \delta =\beta }$ .

3. ^ a b
4. ^ a b Tsalenko 1974.
5. ^ A monomorphism ${\displaystyle \mu :C\to D}$  is said to be strong, if for any epimorphism ${\displaystyle \varepsilon :A\to B}$  and for any morphisms ${\displaystyle \alpha :A\to C}$  and ${\displaystyle \beta :B\to D}$  such that ${\displaystyle \beta \circ \varepsilon =\mu \circ \alpha }$  there exists a morphism ${\displaystyle \delta :B\to C}$ , such that ${\displaystyle \delta \circ \varepsilon =\alpha }$  and ${\displaystyle \mu \circ \delta =\beta }$
6. ^ a b A category ${\displaystyle {\mathcal {K}}}$  is said to be linearly complete, if any functor from a linearly ordered set into ${\displaystyle {\mathcal {K}}}$  has direct and inverse limits.
7. ^ a b A category ${\displaystyle {\mathcal {K}}}$  is said to be well-powered in strong monomorphisms, if for each object ${\displaystyle X}$  the category ${\displaystyle \operatorname {SMono} (X)}$  of all strong monomorphisms into ${\displaystyle X}$  is skeletally small (i.e. has a skeleton which is a set).
8. ^ a b A category ${\displaystyle {\mathcal {K}}}$  is said to be co-well-powered in strong epimorphisms, if for each object ${\displaystyle X}$  the category ${\displaystyle \operatorname {SEpi} (X)}$  of all strong epimorphisms from ${\displaystyle X}$  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 ${\displaystyle {\mathcal {K}}}$ , if each morphism ${\displaystyle \mu }$ , which is not a monomorphism, can be represented as a composition ${\displaystyle \mu =\mu '\circ \varepsilon }$ , where ${\displaystyle \varepsilon }$  is a strong epimorphism which is not an isomorphism.
11. ^ It is said that strong monomorphisms discern epimorphisms in a category ${\displaystyle {\mathcal {K}}}$ , if each morphism ${\displaystyle \varepsilon }$ , which is not an epimorphism, can be represented as a composition ${\displaystyle \varepsilon =\mu \circ \varepsilon '}$ , where ${\displaystyle \mu }$  is a strong monomorphism which is not an isomorphism.
12. ^ Akbarov 2016, p. 31.
13. ^ Akbarov 2016, p. 142.
14. ^ 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.