Nodal decomposition explained

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

\varphi:X\to Y is a representation of \varphi as a product \varphi = \sigma\circ\beta\circ\pi , where \pi is a strong epimorphism,^[1] \beta a bimorphism, and \sigma a strong monomorphism.^[2]

Uniqueness and notations

If it exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions

\varphi = \sigma\circ\beta\circ\pi and \varphi = \sigma'\circ\beta'\circ\pi' there exist isomorphisms η and \theta such that \pi' =η \circ\pi, \beta = \theta\circ\beta'\circ η , \sigma' = \sigma\circ\theta.

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

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

In these notations the nodal decomposition takes the form

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

Connection with the basic decomposition in pre-abelian categories

{ l K} each morphism \varphi has a standard decomposition \varphi = \operatorname{im} \varphi\circ\operatorname{red} \varphi\circ\operatorname{coim} \varphi ,called the basic decomposition (here \operatorname{im} \varphi = \ker(\operatorname{coker} \varphi) , \operatorname{coim} \varphi = \operatorname{coker}(\ker\varphi) , and \operatorname{red} \varphi are respectively the image, the coimage and the reduced part of the morphism \varphi ).

If a morphism

\varphi in a pre-abelian category { l K} has a nodal decomposition, then there exist morphisms η and \theta which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:\operatorname{coim} _{infty} \varphi =η \circ\operatorname{coim} \varphi, \operatorname{red} \varphi = \theta\circ\operatorname{red} _{infty} \varphi\circ η , \operatorname{im} _{infty} \varphi = \operatorname{im} \varphi\circ\theta.

Categories with nodal decomposition

A category

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

{ l K} is linearly complete,^[3] well-powered in strong monomorphisms^[4] and co-well-powered in strong epimorphisms,^[5] then { l K} has nodal decomposition.

More generally, suppose a category

{ l K} is linearly complete, well-powered in strong monomorphisms, co-well-powered in strong epimorphisms, and in addition strong epimorphisms discern monomorphisms^[6] in { l K} , and, dually, strong monomorphisms discern epimorphisms^[7] in { l K} , then { l 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

• Book: Borceux, F.. Handbook of Categorical Algebra 1. Basic Category Theory. registration. Cambridge University Press. 1994. 978-0521061193.
• Book: Tsalenko. M.S.. Shulgeifer. E.G.. Foundations of category theory. Nauka. 1974.
• Akbarov. S.S.. Envelopes and refinements in categories, with applications to functional analysis. Dissertationes Mathematicae. 2016. 513. 1–188. 1110.2013. 10.4064/dm702-12-2015. 118895911.

Notes and References

1. An epimorphism \varepsilon:A\to B is said to be strong, if for any monomorphism \mu:C\to D and for any morphisms \alpha:A\to C and \beta:B\to D such that \beta\circ\varepsilon = \mu\circ\alpha there exists a morphism \delta:B\to C , such that \delta\circ\varepsilon = \alpha and \mu\circ\delta = \beta .
2. A monomorphism \mu:C\to D is said to be strong, if for any epimorphism \varepsilon:A\to B and for any morphisms \alpha:A\to C and \beta:B\to D such that \beta\circ\varepsilon = \mu\circ\alpha there exists a morphism \delta:B\to C , such that \delta\circ\varepsilon = \alpha and \mu\circ\delta = \beta
3. A category { l K} is said to be linearly complete, if any functor from a linearly ordered set into { l K} has direct and inverse limits.
4. A category { l K} is said to be well-powered in strong monomorphisms, if for each object X the category \operatorname{SMono}(X) of all strong monomorphisms into X is skeletally small (i.e. has a skeleton which is a set).
5. A category { l K} is said to be co-well-powered in strong epimorphisms, if for each object X the category \operatorname{SEpi}(X) of all strong epimorphisms from X is skeletally small (i.e. has a skeleton which is a set).
6. It is said that strong epimorphisms discern monomorphisms in a category { l K} , if each morphism \mu , which is not a monomorphism, can be represented as a composition \mu = \mu'\circ\varepsilon , where \varepsilon is a strong epimorphism which is not an isomorphism.
7. It is said that strong monomorphisms discern epimorphisms in a category { l K} , if each morphism \varepsilon , which is not an epimorphism, can be represented as a composition \varepsilon = \mu\circ\varepsilon' , where \mu is a strong monomorphism which is not an isomorphism.