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nonAbelian theory
Definition 0.1.
A nonAbelian 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.
0.1 Examples
ETAC and ETAS axiom interpretations that do not satisfy–in addition to the ETAC or ETAS axioms– the $Ab1$ to $Ab6$ axioms for an abelian category are all examples on nonAbelian categories; a more detailed list is also presented next.
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 $Ab$ every class, or set, of morphisms $Hom_{{Ab}}(,)$ 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 nonAbelian theories it is useful to consider nonAbelian structures so that specific properties determined by the nonAbelian 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 nonAbelian structures.
0.2 Further examples of nonAbelian theories
The following is only a short list of nonAbelian theories:
1. NonAbelian algebraic topology, including also nonAbelian homological algebra; nonAbelian 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. NonAbelian quantum algebraic topology;
3. 4. 5. The axiomatic theory of supercategories (ETAS);
6. 7. $LM_{n}$ Logic algebras;
8. NonAbelian categorical ontology ([3]).
0.3 Remarks
The following alternative definition by Barry Mitchell of an Abelian category should also be mentioned as “an exact additive category with finite products.”.
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):
Theorem 0.1.
“The following statements are equivalent:

(a) $Ab$ is an abelian category;

(b) $Ab$ has kernels, cokernels, finite products, finite coproducts, and is both normal and conormal;

(c) $Ab$ has pushouts and pullbacks and is both normal and conormal.”
References
 1 R. Brown et al. 2008. “NonAbelian Algebraic Topology”. vols. 1 and 2. (Preprint).
 2 R. Brown. 2008. Higher Dimensional Algebra Preprint as pdf and ps docs. at $arXiv:math/0212274v6[math.AT]$
 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: 353408.
Mathematics Subject Classification
1800 no label found18A15 no label found03G12 no label found03G30 no label found03G20 no label found Forums
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