Nodal decomposition

In category theory, an abstract mathematical discipline, a nodal decomposition^[1] 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,^[2][3][4] ${\displaystyle \beta }$ a bimorphism, and ${\displaystyle \sigma }$ a strong monomorphism.^[5][3][4]

Nodal decomposition.

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 ${\displaystyle \varphi =\sigma \circ \beta \circ \pi }$ and ${\displaystyle \varphi =\sigma '\circ \beta '\circ \pi '}$ there exist isomorphisms ${\displaystyle \eta }$ and ${\displaystyle \theta }$ such that

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

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

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

Notations.

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

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

Connection with the basic decomposition in pre-abelian categories

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 ${\displaystyle \varphi }$).

Nodal and basic decompositions.

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:

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

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

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

Categories with nodal decomposition

A category ${\displaystyle {\mathcal {K}}}$ is called a category with nodal decomposition^[1] 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,^[6] well-powered in strong monomorphisms^[7] and co-well-powered in strong epimorphisms,^[8] then ${\displaystyle {\mathcal {K}}}$ has nodal decomposition.^[9]

More generally, suppose a category ${\displaystyle {\mathcal {K}}}$ is linearly complete,^[6] well-powered in strong monomorphisms,^[7] co-well-powered in strong epimorphisms,^[8] and in addition strong epimorphisms discern monomorphisms^[10] in ${\displaystyle {\mathcal {K}}}$, and, dually, strong monomorphisms discern epimorphisms^[11] in ${\displaystyle {\mathcal {K}}}$, then ${\displaystyle {\mathcal {K}}}$ has nodal decomposition.^[12]

The category Ste of stereotype spaces (being non-abelian) has nodal decomposition,^[13] as well as the (non-additive) category SteAlg of stereotype algebras .^[14]