Nodal decomposition

Nodal decomposition.

In category theory, an abstract mathematical discipline, a nodal decompositionTemplate:Sfn of a morphism $\phi :X\to Y$ is a representation of $\phi$ as a product $\phi =\sigma \circ \beta \circ \pi$, where $\pi$ is a strong epimorphism,Template:SfnTemplate:Sfn $\beta$ a bimorphism, and $\sigma$ a strong monomorphism.Template:SfnTemplate:Sfn

Uniqueness and notations

Uniqueness of the nodal decomposition.

If it exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions $\phi =\sigma \circ \beta \circ \pi$ and $\phi ={\sigma }^{\prime }\circ {\beta }^{\prime }\circ {\pi }^{\prime }$ there exist isomorphisms $\eta$ and $\theta$ such that

${\pi }^{\prime }=\eta \circ \pi ,$

$\beta =\theta \circ {\beta }^{\prime }\circ \eta ,$

${\sigma }^{\prime }=\sigma \circ \theta .$

Notations.

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

$\begin{array}{cccc}& \pi ={\mathrm{coim}}_{\mathrm{\infty }}\phi ,& & P={\mathrm{Coim}}_{\mathrm{\infty }}\phi ,\\ & \beta ={\mathrm{red}}_{\mathrm{\infty }}\phi ,& & \\ & \sigma ={\mathrm{im}}_{\mathrm{\infty }}\phi ,& & Q={\mathrm{Im}}_{\mathrm{\infty }}\phi ,\end{array}$

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

In these notations the nodal decomposition takes the form

$\phi ={\mathrm{im}}_{\mathrm{\infty }}\phi \circ {\mathrm{red}}_{\mathrm{\infty }}\phi \circ {\mathrm{coim}}_{\mathrm{\infty }}\phi .$

Connection with the basic decomposition in pre-abelian categories

In a pre-abelian category $𝒦$ each morphism $\phi$ has a standard decomposition

$\phi =\mathrm{im}\phi \circ \mathrm{red}\phi \circ \mathrm{coim}\phi$,

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

Nodal and basic decompositions.

If a morphism $\phi$ in a pre-abelian category $𝒦$ 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:

${\mathrm{coim}}_{\mathrm{\infty }}\phi =\eta \circ \mathrm{coim}\phi ,$

$\mathrm{red}\phi =\theta \circ {\mathrm{red}}_{\mathrm{\infty }}\phi \circ \eta ,$

${\mathrm{im}}_{\mathrm{\infty }}\phi =\mathrm{im}\phi \circ \theta .$

Categories with nodal decomposition

A category $𝒦$ is called a category with nodal decompositionTemplate:Sfn if each morphism $\phi$ has a nodal decomposition in $𝒦$. This property plays an important role in constructing envelopes and refinements in $𝒦$.

In an abelian category $𝒦$ the basic decomposition

$\phi =\mathrm{im}\phi \circ \mathrm{red}\phi \circ \mathrm{coim}\phi$

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

If a pre-abelian category $𝒦$ is linearly complete, well-powered in strong monomorphisms and co-well-powered in strong epimorphisms, then $𝒦$ has nodal decomposition.Template:Sfn

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

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

References

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