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{align} & \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{align} $$ – here $$\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.