00A30-UltracomplexSystems.tex

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\begin{document}
\subsection{Ultra-complex systems}
 
 An \emph{ultra-complex system} represents the human mind from the standpoint
of a generalized Categorical Ontology Theory of Levels as the highest level of complexity that 
emerged through biological and social coevolution over the last $2.2$ million years on Earth.

\subsubsection{Preliminary Data:}

One can represent in square categorical diagrams the emergence of ultra-complex
dynamics from the super-complex dynamics of human organisms coupled {\em via} social interactions 
in characteristic patterns represented by \PMlinkname{Rosetta biogroupoids}{RosettaGroupoids}, together with the complex--albeit inanimate--systems with `chaos'.  With the emergence of the ultra-complex system of the human mind-- based on the super-complex human organism-- there is always an associated
progression towards higher dimensional algebras from the lower
dimensions of human neural network dynamics and the simple algebra
of physical dynamics, as shown in the following, essentially \emph{non-commutative} categorical diagram.

\begin{definition} 
An \emph{ultra-complex system, $U_{CS}$} is defined as an object representation in the following non-commutative
diagram of systems and dynamic system morphisms or `dynamic transformations':

 $$ \xymatrix@C=5pc{[SUPER-COMPLEX] \ar [r] ^{(\textbf{Higher
Dim})} \ar[d] _{\Lambda}& ~~~(U_{CS}= ULTRA-COMPLEX) \ar [d]^{onto}\\ COMPLEX&
\ar [l] ^{(\textbf{Generic Map})}[SIMPLE]} $$
\end{definition}
Note that the above diagram is indeed not `natural' (i.e. it is not commutative) for reasons
related to the emergence of the higher dimensions of the super--complex
(biological/organismic) and/or ultra--complex (psychological/neural network dynamic) levels in comparison with
the low dimensions of either simple (physical/classical) or complex (chaotic) dynamic systems. 

 
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\end{document}