In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism is a representation of as a product , where is a strong epimorphism, a bimorphism, and a strong monomorphism.
Uniqueness and notations
If exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions and there exist isomorphisms and such that
This property justifies some special notations for the elements of the nodal decomposition:
– here and are called the nodal coimage of , and the nodal image of , and the nodal reduced part of .
In these notations the nodal decomposition takes the form
Connection with the basic decomposition in pre-abelian categories
In a pre-abelian category each morphism has a standard decomposition
called the basic decomposition (here , , and are respectively the image, the coimage and the reduced part of the morphism ).
If a morphism in a pre-abelian category has a nodal decomposition, then there exist morphisms and which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:
Categories with nodal decomposition
In an abelian category the basic decomposition
is always nodal. As a corollary, all abelian categories have 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 monomorphisms in , and, dually, strong monomorphisms discern epimorphisms in , then has nodal decomposition.
- Akbarov 2016, p. 28.
- An epimorphism is said to be strong, if for any monomorphism and for any morphisms and such that there exists a morphism , such that and .
- Borceux 1994.
- Tsalenko 1974. sfn error: no target: CITEREFTsalenko1974 (help)
- A monomorphism is said to be strong, if for any epimorphism and for any morphisms and such that there exists a morphism , such that and
- A category is said to be linearly complete, if any functor from a linearly ordered set into has direct and inverse limits.
- A category is said to be well-powered in strong monomorphisms, if for each object the category of all strong monomorphisms into is skeletally small (i.e. has a skeleton which is a set).
- A category is said to be co-well-powered in strong epimorphisms, if for each object the category of all strong epimorphisms from is skeletally small (i.e. has a skeleton which is a set).
- Akbarov 2016, p. 37.
- It is said that strong epimorphisms discern monomorphisms in a category , if each morphism , which is not a monomorphism, can be represented as a composition , where is a strong epimorphism which is not an isomorphism.
- It is said that strong monomorphisms discern epimorphisms in a category , if each morphism , which is not an epimorphism, can be represented as a composition , where is a strong monomorphism which is not an isomorphism.
- Akbarov 2016, p. 31.
- Akbarov 2016, p. 142.
- Akbarov 2016, p. 164.
- 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.