Nodal decomposition

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]

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]

Notes

1. Akbarov 2016, p. 28.
2. An epimorphism ${\displaystyle \varepsilon :A\to B}$ is said to be strong, if for any monomorphism ${\displaystyle \mu :C\to D}$ and for any morphisms ${\displaystyle \alpha :A\to C}$ and ${\displaystyle \beta :B\to D}$ such that ${\displaystyle \beta \circ \varepsilon =\mu \circ \alpha }$ there exists a morphism ${\displaystyle \delta :B\to C}$, such that ${\displaystyle \delta \circ \varepsilon =\alpha }$ and ${\displaystyle \mu \circ \delta =\beta }$.

   

3. Borceux 1994.
4. Tsalenko & Shulgeifer 1974.
5. A monomorphism ${\displaystyle \mu :C\to D}$ is said to be strong, if for any epimorphism ${\displaystyle \varepsilon :A\to B}$ and for any morphisms ${\displaystyle \alpha :A\to C}$ and ${\displaystyle \beta :B\to D}$ such that ${\displaystyle \beta \circ \varepsilon =\mu \circ \alpha }$ there exists a morphism ${\displaystyle \delta :B\to C}$, such that ${\displaystyle \delta \circ \varepsilon =\alpha }$ and ${\displaystyle \mu \circ \delta =\beta }$
6. A category ${\displaystyle {\mathcal {K}}}$ is said to be linearly complete, if any functor from a linearly ordered set into ${\displaystyle {\mathcal {K}}}$ has direct and inverse limits.
7. A category ${\displaystyle {\mathcal {K}}}$ is said to be well-powered in strong monomorphisms, if for each object ${\displaystyle X}$ the category ${\displaystyle \operatorname {SMono} (X)}$ of all strong monomorphisms into ${\displaystyle X}$ is skeletally small (i.e. has a skeleton which is a set).
8. A category ${\displaystyle {\mathcal {K}}}$ is said to be co-well-powered in strong epimorphisms, if for each object ${\displaystyle X}$ the category ${\displaystyle \operatorname {SEpi} (X)}$ of all strong epimorphisms from ${\displaystyle X}$ is skeletally small (i.e. has a skeleton which is a set).
9. Akbarov 2016, p. 37.
10. It is said that strong epimorphisms discern monomorphisms in a category ${\displaystyle {\mathcal {K}}}$, if each morphism ${\displaystyle \mu }$, which is not a monomorphism, can be represented as a composition ${\displaystyle \mu =\mu '\circ \varepsilon }$, where ${\displaystyle \varepsilon }$ is a strong epimorphism which is not an isomorphism.
11. It is said that strong monomorphisms discern epimorphisms in a category ${\displaystyle {\mathcal {K}}}$, if each morphism ${\displaystyle \varepsilon }$, which is not an epimorphism, can be represented as a composition ${\displaystyle \varepsilon =\mu \circ \varepsilon '}$, where ${\displaystyle \mu }$ is a strong monomorphism which is not an isomorphism.
12. Akbarov 2016, p. 31.
13. Akbarov 2016, p. 142.
14. Akbarov 2016, p. 164.

References

• 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.