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category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, 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 In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
, and \sigma a strong monomorphism.A
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
\mu:C\to D is said to be strong, if for any
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
\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
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
s \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 In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enrich ...
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 In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enrich ...
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
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sho ...
s and refinements in . In an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of a ...
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 In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enrich ...
is linearly complete, A category is said to be ''linearly complete'', if any functor from a linearly ordered set into has
direct Direct may refer to: Mathematics * Directed set, in order theory * Direct limit of (pre), sheaves * Direct sum of modules, a construction in abstract algebra which combines several vector spaces Computing * Direct access (disambiguation), a ...
and
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits c ...
s.
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 Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-functionn see Sigma additivity * Additive category, a preadditive category with fi ...
) category SteAlg of
stereotype algebra In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense. Definition A topological algebra A over a topological field K is a ...
s .


Notes


References

* * *{{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