Examples

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.

Further examples of non-Abelian theories

The following is only a short list of non-Abelian theories:

1. non-Abelian algebraic topology, including also non-Abelian homological algebra; non-Abelian algebraic topology overview and R. Brown 2008 preprint, ([1,2]).
   (See also the recent book exposition with the title “Nonabelian Algebraic Topology” vol. 1 by Brown and Sivera,(respectively, vol. 2 with Higgins, in preparation).
2. Non-Abelian Quantum Algebraic Topology;
3. Non-Abelian gauge field theory (in Quantum Physics);
4. noncommutative geometry;
5. The axiomatic theory of supercategories (ETAS);
6. higher dimensional algebra (HDA)
7. Logic algebras;
8. Non-Abelian categorical ontology ([3]).

Remarks

He also published in his textbook the following theorem: (Theorem 20.1, on p.33 of Barry Mitchell in “Theory of Catgeories”, 1965, Academic Press: New York and London):

Bibliography

1

R. Brown et al. 2008. “Non-Abelian Algebraic Topology”. vols. 1 and 2. (Preprint).

2

R. Brown. 2008. Higher Dimensional Algebra Preprint as pdf and ps docs. at

3

I. C. Baianu, R. Brown and J. F. Glazebrook. 2007, A Non–Abelian Categorical Ontology and Higher Dimensional Algebra of Spacetimes and Quantum Gravity., Axiomathes, 17: 353-408.