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,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 . a bimorphism, and a strong monomorphism.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
Uniqueness and notations If it 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 A category is called a category with nodal decomposition if each morphism has a nodal decomposition in . This property plays an important role in constructing envelopes and refinements in .
In an abelian category the basic decomposition
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 the category of all strong monomorphisms into 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 the category of all strong epimorphisms from 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 , which is not a monomorphism, can be represented as a composition , where 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 , which is not an epimorphism, can be represented as a composition , where 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 .
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