# Nodal decomposition

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.

## Uniqueness and notations

If it 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
    $\pi '=\eta \circ \pi ,$
    $\beta =\theta \circ \beta '\circ \eta ,$
    $\sigma '=\sigma \circ \theta .$
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 $\varphi$).
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 .