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non-Abelian theory

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\begin{document}
\begin{definition}
A {\em 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. 
\end{definition}

\subsection{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 \PMlinkname{abelian category}{AbelianCategory} are all examples on non-Abelian categories; a more detailed list is also presented next.

\begin{remark}
 In a general sense, any Abelian category (or {\em 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 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 {\em non-Abelian structures}. 
\end{remark}

\subsection{Further examples of non-Abelian theories}

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

\begin{enumerate}

\item Non-Abelian algebraic topology, including also non-Abelian homological algebra;
\PMlinkexternal{non-Abelian algebraic topology overview}{http://www.bangor.ac.uk/~mas010/nonab-a-t.html} and
\PMlinkexternal{R. Brown 2008 preprint}{http://arxiv.org/abs/math/0212274}, (\cite{RBetal2k7,RB2k8}).\\
(See also the \PMlinkexternal{recent book exposition}{2008 http://planetmath.org/?op=getobj&from=lec&id=75} with the title {\em ``Nonabelian Algebraic Topology''} vol. 1 by Brown and Sivera,(respectively, vol. 2 with Higgins, \emph{in preparation}). 

\item Non-Abelian quantum algebraic topology;

\item Non-Abelian gauge field theory (in Quantum Physics);

\item Noncommutative geometry;

\item The axiomatic theory of supercategories (ETAS); 

\item Higher dimensional algebra (HDA)
\item $LM_n$ Logic algebras;

\item Non-Abelian categorical ontology (\cite{BBG2k7}). 

\end{enumerate}

\subsection{Remarks}
 The following alternative definition by Barry Mitchell of an Abelian category should also be mentioned as {\em ``an exact additive category with finite products.''}. 

 He also published in his textbook the following theorem:
(\textbf{Theorem 20.1}, on p.33 of Barry Mitchell in ``Theory of Catgeories'', 1965, Academic Press: 
New York and London): 

\begin{theorem} 
 ``{\em The following statements are equivalent}:
\begin{itemize}
\item (a) $Ab$ is an abelian category; 
\item (b) $Ab$ has kernels, cokernels, finite products, finite coproducts,
 and is both normal and conormal; 
\item (c) $Ab$ has pushouts and pullbacks and is both normal and conormal.''
\end{itemize}
\end{theorem}

\begin{thebibliography}{9}

\bibitem{RBetal2k7}
R. Brown et al. 2008. {\em ``Non-Abelian Algebraic Topology''}. vols. 1 and 2. ({\em Preprint}).

\bibitem{RB2k8}
R. Brown. 2008. {\em Higher Dimensional Algebra Preprint as pdf and ps docs. at $arXiv:math/0212274v6 [math.AT]$}

\bibitem{BBG2k7}
I. C. Baianu, R. Brown and J. F. Glazebrook. 2007,  A Non--Abelian Categorical Ontology and Higher Dimensional Algebra of Spacetimes and Quantum Gravity., {\em Axiomathes }, \textbf{17}: 353-408. 


\end{thebibliography}
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