Nodal decomposition

In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism ${\displaystyle \varphi :X\to Y}$is a representation of ${\displaystyle \varphi }$as a product ${\displaystyle \varphi =\sigma \circ \beta \circ \pi }$, where ${\displaystyle \pi }$is a strong epimorphism, ${\displaystyle \beta }$a bimorphism, and ${\displaystyle \sigma }$a strong monomorphism.

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

${\displaystyle \pi '=\eta \circ \pi ,}$

${\displaystyle \beta =\theta \circ \beta '\circ \eta ,}$

${\displaystyle \sigma '=\sigma \circ \theta .}$

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

${\displaystyle {\begin{aligned}&\pi =\operatorname {coim} _{\infty }\varphi ,&&P=\operatorname {Coim} _{\infty }\varphi ,\\&\beta =\operatorname {red} _{\infty }\varphi ,&&\\&\sigma =\operatorname {im} _{\infty }\varphi ,&&Q=\operatorname {Im} _{\infty }\varphi ,\end{aligned}}}$

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

In these notations the nodal decomposition takes the form

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

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

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

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

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

${\displaystyle \operatorname {coim} _{\infty }\varphi =\eta \circ \operatorname {coim} \varphi ,}$

${\displaystyle \operatorname {red} \varphi =\theta \circ \operatorname {red} _{\infty }\varphi \circ \eta ,}$

${\displaystyle \operatorname {im} _{\infty }\varphi =\operatorname {im} \varphi \circ \theta .}$

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

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

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

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

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

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

• Borceux, F. (1994). Handbook of Categorical Algebra 1. Basic Category Theory. Cambridge University Press. ISBN 978-0521061193.
• 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. arXiv:1110.2013. doi:10.4064/dm702-12-2015. S2CID 118895911.