Nodal decomposition

File:Definition-of-a-nodal-decomposition.jpg In category theory, an abstract mathematical discipline, a nodal decomposition{{sfn|Akbarov|2016|p=28}} of a morphism $\backslash varphi:X\backslash to\; Y$ is a representation of $\backslash varphi$ as a product $\backslash varphi=\backslash sigma\backslash circ\backslash beta\backslash circ\backslash pi$, where $\backslash pi$ is a strong epimorphism,An epimorphism $\backslash varepsilon:A\backslash to\; B$ is said to be strong, if for any monomorphism $\backslash mu:C\backslash to\; D$ and for any morphisms $\backslash alpha:A\backslash to\; C$ and $\backslash beta:B\backslash to\; D$ such that $\backslash beta\backslash circ\backslash varepsilon=\backslash mu\backslash circ\backslash alpha$ there exists a morphism $\backslash delta:B\backslash to\; C$, such that $\backslash delta\backslash circ\backslash varepsilon=\backslash alpha$ and $\backslash mu\backslash circ\backslash delta=\backslash beta$. thumb{{sfn|Borceux|1994}}{{sfn|Tsalenko|Shulgeifer|1974}} $\backslash beta$ a bimorphism, and $\backslash sigma$ a strong monomorphism.A monomorphism $\backslash mu:C\backslash to\; D$ is said to be strong, if for any epimorphism $\backslash varepsilon:A\backslash to\; B$ and for any morphisms $\backslash alpha:A\backslash to\; C$ and $\backslash beta:B\backslash to\; D$ such that $\backslash beta\backslash circ\backslash varepsilon=\backslash mu\backslash circ\backslash alpha$ there exists a morphism $\backslash delta:B\backslash to\; C$, such that $\backslash delta\backslash circ\backslash varepsilon=\backslash alpha$ and $\backslash mu\backslash circ\backslash delta=\backslash beta${{sfn|Borceux|1994}}{{sfn|Tsalenko|Shulgeifer|1974}}

== Uniqueness and notations ==

File:Uniqueness-of-nodal-decomposition-2.jpg If it exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions $\backslash varphi=\backslash sigma\backslash circ\backslash beta\backslash circ\backslash pi$ and $\backslash varphi=\backslash sigma\text{'}\backslash circ\backslash beta\text{'}\backslash circ\backslash pi\text{'}$ there exist isomorphisms $\backslash eta$ and $\backslash theta$ such that

: $\backslash pi\text{'}=\backslash eta\backslash circ\backslash pi,$

: $\backslash beta=\backslash theta\backslash circ\backslash beta\text{'}\backslash circ\backslash eta,$

: $\backslash sigma\text{'}=\backslash sigma\backslash circ\backslash theta.$

File:Nodal-decomposition-notations.jpg

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

In these notations the nodal decomposition takes the form

:$\backslash varphi=\backslash operatorname\{im\}\_\backslash infty\; \backslash varphi\backslash circ\backslash operatorname\{red\}\_\backslash infty\; \backslash varphi\; \backslash circ\; \backslash operatorname\{coim\}\_\backslash infty\; \backslash varphi.$

== Connection with the basic decomposition in pre-abelian categories ==

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

: $\backslash varphi=\backslash operatorname\{im\}\; \backslash varphi\backslash circ\backslash operatorname\{red\}\; \backslash varphi\backslash circ\backslash operatorname\{coim\}\; \backslash varphi$,

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

File:Nodal-and-basic-decomposition-1.jpg If a morphism $\backslash varphi$ in a pre-abelian category $\{\backslash mathcal\; K\}$ has a nodal decomposition, then there exist morphisms $\backslash eta$ and $\backslash theta$ which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:

: $\backslash operatorname\{coim\}\_\backslash infty\; \backslash varphi=\backslash eta\backslash circ\backslash operatorname\{coim\}\; \backslash varphi,$

: $\backslash operatorname\{red\}\; \backslash varphi=\backslash theta\backslash circ\backslash operatorname\{red\}\_\backslash infty\; \backslash varphi\backslash circ\backslash eta,$

: $\backslash operatorname\{im\}\_\backslash infty\; \backslash varphi=\backslash operatorname\{im\}\; \backslash varphi\backslash circ\backslash theta.$

== Categories with nodal decomposition ==

A category $\{\backslash mathcal\; K\}$ is called a category with nodal decomposition{{sfn|Akbarov|2016|p=28}} if each morphism $\backslash varphi$ has a nodal decomposition in $\{\backslash mathcal\; K\}$. This property plays an important role in constructing envelopes and refinements in $\{\backslash mathcal\; K\}$.

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

: $\backslash varphi=\backslash operatorname\{im\}\; \backslash varphi\backslash circ\backslash operatorname\{red\}\; \backslash varphi\backslash circ\backslash operatorname\{coim\}\; \backslash varphi$

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

If a pre-abelian category $\{\backslash mathcal\; K\}$ is linearly complete, A category $\{\backslash mathcal\; K\}$ is said to be linearly complete, if any functor from a linearly ordered set into $\{\backslash mathcal\; K\}$ has direct and inverse limits. well-powered in strong monomorphismsA category $\{\backslash mathcal\; K\}$ is said to be well-powered in strong monomorphisms, if for each object $X$ the category $\backslash operatorname\{SMono\}(X)$ 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 $\{\backslash mathcal\; K\}$ is said to be co-well-powered in strong epimorphisms, if for each object $X$ the category $\backslash operatorname\{SEpi\}(X)$ of all strong epimorphisms from $X$ is skeletally small (i.e. has a skeleton which is a set). then $\{\backslash mathcal\; K\}$ has nodal decomposition.{{sfn|Akbarov|2016|p=37}}

More generally, suppose a category $\{\backslash mathcal\; K\}$ 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 $\{\backslash mathcal\; K\}$, if each morphism $\backslash mu$, which is not a monomorphism, can be represented as a composition $\backslash mu=\backslash mu\text{'}\backslash circ\backslash varepsilon$, where $\backslash varepsilon$ is a strong epimorphism which is not an isomorphism. in $\{\backslash mathcal\; K\}$, and, dually, strong monomorphisms discern epimorphismsIt is said that strong monomorphisms discern epimorphisms in a category $\{\backslash mathcal\; K\}$, if each morphism $\backslash varepsilon$, which is not an epimorphism, can be represented as a composition $\backslash varepsilon=\backslash mu\backslash circ\backslash varepsilon\text{'}$, where $\backslash mu$ is a strong monomorphism which is not an isomorphism. in $\{\backslash mathcal\; K\}$, then $\{\backslash mathcal\; K\}$ has nodal decomposition.{{sfn|Akbarov|2016|p=31}}

The category Ste of stereotype spaces (being non-abelian) has nodal decomposition,{{sfn|Akbarov|2016|p=142}} as well as the (non-additive) category SteAlg of stereotype algebras .{{sfn|Akbarov|2016|p=164}}