Definition 0.1
A
non-Abelian theory is one that does not satisfy one, several, or all of the axioms of an
Abelian theory, such as, for example, those for an
Abelian category
theory.
Remark 0.1
In a general sense, any Abelian category (or
abelian category) can be regarded as a `good' model for the
category
of Abelian, or commutative, groups. Furthermore, in an Abelian category

every class, or set, of morphisms

forms an Abelian (or commutative) group. There are several strict definitions of Abelian
categories involving 3, 4 or up to 6 axioms defining the Abelian character of a category.
To illustrate non-Abelian theories it is useful to consider non-Abelian structures so that
specific properties determined by the non-Abelian set of axioms become `transparent' in terms
of the properties of objects for example for concrete categories that have objects; such examples
are presented separately as
non-Abelian structures.