Nodal decomposition
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). 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 .