Nodal decomposition

In category theory, an abstract mathematical discipline, a nodal decomposition[1] of a morphism

φ = σ ∘ β ∘ π

is a strong epimorphism,[2][3][4]

[5][3][4] If it exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions

φ = σ ∘ β ∘ π

such that This property justifies some special notations for the elements of the nodal decomposition: – here

are called the nodal coimage of

the nodal reduced part of

In these notations the nodal decomposition takes the form In a pre-abelian category

has a standard decomposition called the basic decomposition (here

im ⁡ φ = ker ⁡ ( coker ⁡ φ )

coim ⁡ φ = coker ⁡ ( ker ⁡ φ )

are respectively the image, the coimage and the reduced part of the morphism

has a nodal decomposition, then there exist morphisms

which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities: A category

is called a category with nodal decomposition[1] 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.

is linearly complete,[6] well-powered in strong monomorphisms[7] and co-well-powered in strong epimorphisms,[8] then

[9] More generally, suppose a category

is linearly complete,[6] well-powered in strong monomorphisms,[7] co-well-powered in strong epimorphisms,[8] and in addition strong epimorphisms discern monomorphisms[10] in

, and, dually, strong monomorphisms discern epimorphisms[11] in

[12] The category Ste of stereotype spaces (being non-abelian) has nodal decomposition,[13] as well as the (non-additive) category SteAlg of stereotype algebras .