Nodal decomposition

 In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism formula_1 is a representation of formula_2 as a product formula_3, where formula_4 is a strong epimorphism, formula_5 a bimorphism, and formula_6 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 formula_3 and formula_8 there exist isomorphisms formula_9 and formula_10 such that 
This property justifies some special notations for the elements of the nodal decomposition: 
– here formula_15 and formula_16 are called the "nodal coimage of formula_2", formula_18 and formula_19 the "nodal image of formula_2", and formula_21 the "nodal reduced part of formula_2".
In these notations the nodal decomposition takes the form
Connection with the basic decomposition in pre-abelian categories.
In a pre-abelian category formula_24 each morphism formula_2 has a standard decomposition 
called the "basic decomposition" (here formula_27, formula_28, and formula_29 are respectively the image, the coimage and the reduced part of the morphism formula_2).
 If a morphism formula_2 in a pre-abelian category formula_24 has a nodal decomposition, then there exist morphisms formula_9 and formula_10 which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:
Categories with nodal decomposition.
A category formula_24 is called a "category with nodal decomposition" if each morphism formula_2 has a nodal decomposition in formula_24. This property plays an important role in constructing envelopes and refinements in formula_24. 
In an abelian category formula_24 the basic decomposition
is always nodal. As a corollary, "all abelian categories have nodal decomposition". 
"If a pre-abelian category formula_24 is linearly complete, well-powered in strong monomorphisms and co-well-powered in strong epimorphisms, then formula_24 has nodal decomposition."
More generally, "suppose a category formula_24 is linearly complete, well-powered in strong monomorphisms, co-well-powered in strong epimorphisms, and in addition strong epimorphisms discern monomorphisms in formula_24, and, dually, strong monomorphisms discern epimorphisms in formula_24, then formula_24 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 .