%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: ultracomplex systems and categorical dynamics
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%%% Owner: bci1
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\begin{document}
\section{Categorical Dynamics}
A relatively recent area (1958- ) of \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} and \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} developing applications of \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html}, \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}, higher dimensional categories and \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html} to \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} in classical, quantum, complex and super-complex \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}.
One can represent in \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} \htmladdnormallink{categorical diagrams}{http://planetphysics.us/encyclopedia/CategoricalDiagramsDefinedByFunctors.html} the emergence of ultra-complex
dynamics from the super-complex dynamics of human organisms coupled {\em via} social interactions in characteristic patterns represented by Rosetta biogroupoids, 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{\htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}} categorical diagram.
\begin{definition}
An \emph{ultra-complex system, $U_{CS}$} is defined as an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} in the following \htmladdnormallink{non-commutative diagram}{http://planetphysics.us/encyclopedia/NonCommutativeDynamicModelingDiagrams.html} of systems and \htmladdnormallink{dynamic system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} or `dynamic transformations':
$$ \xymatrix@C=5pc{[SUPER-COMPLEX] \ar [r] ^{(\textbf{Higher
Dim})} \ar[d] _{\Lambda}& [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]])(U_{CS}= ULTRA-COMPLEX) \ar [d]^{onto}\\ COMPLEX&
\ar [l] ^{(\textbf{Generic Map})}[SIMPLE]} $$
\end{definition}
Note that the above \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} 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.
\end{document}
