Nodal decomposition

Nodal decomposition.

In category theory, an abstract mathematical discipline, a nodal decomposition 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,

$\beta$

a bimorphism, and

$\sigma$

a strong monomorphism.

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:

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

– 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 decomposition 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.

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.

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

References

• Borceux, F. (1994). Handbook of Categorical Algebra 1. Basic Category Theory. Cambridge University Press. ISBN 978-0521061193. https://archive.org/details/handbookofcatego0000borc. 
• 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. doi:10.4064/dm702-12-2015. https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513.