Talk:PlanetPhysics/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 \htmladdnormallink{Abelian theory}{http://planetphysics.us/encyclopedia/AbelianCategory.html}, such as, for example, those for an \htmladdnormallink{Abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory.html} theory.
\end{definition}

\subsection{Examples}
\htmladdnormallink{ETAC}{http://planetphysics.us/encyclopedia/ETACAxioms.html} and \htmladdnormallink{ETAS axiom}{http://planetphysics.us/encyclopedia/ETACAxioms.html} interpretations that do not satisfy--in addition to the ETAC or \htmladdnormallink{ETAS}{http://planetphysics.us/encyclopedia/ETACAxioms.html} axioms-- the $Ab1$ to $Ab6$ axioms for an \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory.html} 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 \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of Abelian, or commutative, \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. Furthermore, in an Abelian category $Ab$ every class, or set, of \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $Hom_{Ab}(-,-)$ 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 \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} 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 \htmladdnormallink{non-Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html}, including also non-Abelian homological algebra;
\htmladdnormallink{non-Abelian algebraic topology overview}{http://www.bangor.ac.uk/~mas010/nonab-a-t.html} and
\htmladdnormallink{R. Brown 2008 preprint}{http://arxiv.org/abs/math/0212274}, (\cite{RBetal2k7,RB2k8}).\\
(See also the \htmladdnormallink{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 \htmladdnormallink{Non-Abelian Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html};

\item Non-Abelian gauge \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} theory (in Quantum Physics);

\item \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html};

\item The axiomatic theory of \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html} (ETAS);

\item \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html})
\item $LM_n$ Logic algebras;

\item Non-Abelian \htmladdnormallink{categorical ontology}{http://planetphysics.us/encyclopedia/CategoricalOntology.html} (\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 \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}:
(\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} 

\end{document}