In category theory, an abstract mathematical discipline, a nodal decomposition [1] of a morphism is a representation of as a product , where is a strong epimorphism, [2] [3] [4] a bimorphism, and a strong monomorphism. [5] [3] [4]
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
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:
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.
If a pre-abelian category is linearly complete, [6] well-powered in strong monomorphisms [7] and co-well-powered in strong epimorphisms, [8] then has nodal decomposition. [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 , then has nodal decomposition. [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 . [14]
In category theory, an abstract mathematical discipline, a nodal decomposition [1] of a morphism is a representation of as a product , where is a strong epimorphism, [2] [3] [4] a bimorphism, and a strong monomorphism. [5] [3] [4]
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
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:
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.
If a pre-abelian category is linearly complete, [6] well-powered in strong monomorphisms [7] and co-well-powered in strong epimorphisms, [8] then has nodal decomposition. [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 , then has nodal decomposition. [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 . [14]