PlanetPhysics/Non Abelian Theory

A non-Abelian theory is one that does not satisfy one, several, or all of the axioms of an [[../AbelianCategory/|Abelian theory]], such as, for example, those for an [[../AbelianCategory/|Abelian category]] theory.

Examples

[[../ETACAxioms/|ETAC]] and [[../ETACAxioms/|ETAS axiom]] interpretations that do not satisfy--in addition to the ETAC or [[../ETACAxioms/|ETAS]] axioms-- the $Ab1$ to $Ab6$ axioms for an [[../AbelianCategory/|abelian category]] are all examples on non-Abelian categories; a more detailed list is also presented next.

In a general sense, any Abelian category (or abelian category ) can be regarded as a `good' model for the [[../Cod/|category]] of Abelian, or commutative, [[../TrivialGroupoid/|groups]]. Furthermore, in an Abelian category $Ab$ every class, or set, of [[../TrivialGroupoid/|morphisms]] $Ho{m}_{Ab}(-,-)$ 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 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 [[../TrivialGroupoid/|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. [[../ModuleAlgebraic/|non-Abelian algebraic topology]], including also non-Abelian homological algebra;

non-Abelian algebraic topology overview and R. Brown 2008 preprint, ().\\ (See also the [2008 http://planetmath.org/?op=getobj&from=lec&id=75 recent book exposition] with the title "Nonabelian Algebraic Topology" vol. 1 by Brown and Sivera,(respectively, vol. 2 with Higgins, in preparation ).

1. [[../NonAbelianQuantumAlgebraicTopology3/|Non-Abelian Quantum Algebraic Topology]];

1. Non-Abelian gauge [[../CosmologicalConstant2/|field]] theory (in Quantum Physics);

1. [[../NoncommutativeGeometry4/|noncommutative geometry]];

1. The axiomatic theory of [[../SuperCategory6/|supercategories]] (ETAS);

1. [[../HigherDimensionalAlgebra2/|higher dimensional algebra]] ([[../2Groupoid2/|HDA]])
2. $L{M}_n$ Logic algebras;

1. Non-Abelian [[../CategoricalOntology/|categorical ontology]] ().

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 [[../Formula/|theorem]]: (Theorem 20.1 , on p.33 of Barry Mitchell in "Theory of Catgeories", 1965, Academic Press: New York and London):

\begin{theorem} 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.

\end{theorem}