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## Summary

In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism $\varphi :X\to Y$ is a representation of $\varphi$ as a product $\varphi =\sigma \circ \beta \circ \pi$ , where $\pi$ is a strong epimorphism, $\beta$ a bimorphism, and $\sigma$ a strong monomorphism. 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 $\varphi =\sigma \circ \beta \circ \pi$  and $\varphi =\sigma '\circ \beta '\circ \pi '$  there exist isomorphisms $\eta$  and $\theta$  such that

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

Notations.

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

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

In these notations the nodal decomposition takes the form

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

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

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

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

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

Nodal and basic decompositions.

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

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

## Categories with nodal decomposition

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

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

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

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

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

More generally, suppose a category ${\mathcal {K}}$  is linearly complete, well-powered in strong monomorphisms, co-well-powered in strong epimorphisms, and in addition strong epimorphisms discern monomorphisms in ${\mathcal {K}}$ , and, dually, strong monomorphisms discern epimorphisms in ${\mathcal {K}}$ , then ${\mathcal {K}}$  has nodal decomposition.

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