Nodal decomposition

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, $\beta$ a bimorphism, and $\sigma$ a strong monomorphism.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 $\mu\circ\delta=\beta$

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 $\eta$ and $\theta$ such that
: $\pi'=\eta\circ\pi,$
: $\beta=\theta\circ\beta'\circ\eta,$
: $\sigma'=\sigma\circ\theta.$

This property justifies some special notations for the elements of the nodal decomposition:
:$\begin & \pi=\operatorname_\infty \varphi, && P=\operatorname_\infty \varphi,\\ & \beta=\operatorname_\infty \varphi, && \\ & \sigma=\operatorname_\infty \varphi, && Q=\operatorname_\infty \varphi, \end$
– here $\operatorname_\infty \varphi$ and $\operatorname_\infty \varphi$ are called the ''nodal coimage of $\varphi$'', $\operatorname_\infty \varphi$ and $\operatorname_\infty \varphi$ the ''nodal image of $\varphi$'', and $\operatorname_\infty \varphi$ the ''nodal reduced part of $\varphi$''.

In these notations the nodal decomposition takes the form
:$\varphi=\operatorname_\infty \varphi\circ\operatorname_\infty \varphi \circ \operatorname_\infty \varphi.$

Connection with the basic decomposition in pre-abelian categories

In a pre-abelian category $$ each morphism $\varphi$ has a standard decomposition
: $\varphi=\operatorname \varphi\circ\operatorname \varphi\circ\operatorname \varphi$,
called the ''basic decomposition'' (here $\operatorname \varphi=\ker(\operatorname \varphi)$, $\operatorname \varphi=\operatorname(\ker\varphi)$, and $\operatorname \varphi$ are respectively the image, the coimage and the reduced part of the morphism $\varphi$).

If a morphism $\varphi$ in a pre-abelian category $$ 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:
: $\operatorname_\infty \varphi=\eta\circ\operatorname \varphi,$
: $\operatorname \varphi=\theta\circ\operatorname_\infty \varphi\circ\eta,$
: $\operatorname_\infty \varphi=\operatorname \varphi\circ\theta.$

Categories with nodal decomposition

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

In an abelian category $$ the basic decomposition
: $\varphi=\operatorname \varphi\circ\operatorname \varphi\circ\operatorname \varphi$
is always nodal. As a corollary, ''all abelian categories have nodal decomposition''.

''If a pre-abelian category $$ is linearly complete, A category $$ is said to be ''linearly complete'', if any functor from a linearly ordered set into $$ has direct and inverse limits. well-powered in strong monomorphismsA category $$ is said to be ''well-powered in strong monomorphisms'', if for each object $X$ the category $\operatorname(X)$ of all strong monomorphisms into $X$ is skeletally small (i.e. has a skeleton which is a set). and co-well-powered in strong epimorphisms,A category $$ is said to be ''co-well-powered in strong epimorphisms'', if for each object $X$ the category $\operatorname(X)$ of all strong epimorphisms from $X$ is skeletally small (i.e. has a skeleton which is a set). then $$ has nodal decomposition.''

More generally, ''suppose a category $$ is linearly complete, well-powered in strong monomorphisms, co-well-powered in strong epimorphisms, and in addition strong epimorphisms discern monomorphismsIt is said that ''strong epimorphisms discern monomorphisms'' in a category $$, 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. in $$, and, dually, strong monomorphisms discern epimorphismsIt is said that ''strong monomorphisms discern epimorphisms'' in a category $$, 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. in $$, then $$ 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 .

Notes

References

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*{{cite journal, last=Akbarov, first=S.S., title=Envelopes and refinements in categories, with applications to functional analysis, url=https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513, journal=Dissertationes Mathematicae, year=2016, volume=513, pages=1–188, arxiv=1110.2013, doi=10.4064/dm702-12-2015, s2cid=118895911

Category theory