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categorical dynamics

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\begin{document}
\subsection{Introduction}

 Categorical dynamics is a relatively recent area (1958- ) of applied algebraic topology/category theory and higher dimensional algebra concerned with system dynamics that utilizes concepts such as: categories, functors, natural transformations, higher dimensional categories and \PMlinkname{supercategories}{Supercategory} to study motion and dynamic processes in classical/ quantum systems, as well as complex or super-complex systems (biodynamics).
 
 A type of categorical dynamics was first introduced and studied by William F. Lawvere for classical systems. Subsequently, a complex class of categorical, dynamic $(M,R)$--systems representing the categorical dynamics involved in metabolic--replication processes in terms of categories of sets and ODE's was reported by Robert Rosen in 1970.

 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 non-commutative categorical diagram of dynamic systems and their transformations.

\subsection{Basic definitions in categorical dynamics}

\begin{definition}
An \emph{ultra-complex system, $U_{CS}$} is defined as an object representation in the following non-commutative
diagram of dynamic 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}

 One notes that the above diagram is indeed not `natural' (that is, 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. Moreover,
each type of dynamic system shown in the above diagram is in its turn represented by a distinct diagram
representing its dynamics in terms of transitions occurring in a state space $S$ according to one or several
transition functions or dynamic laws, denoted by $\delta$ for either classical or chaotic physical systems and
by a class of transition functions: 
$$\left\{\lambda_{\tau} \right\} _{\tau \in T},$$

where $T$ is an index class consisting of dynamic parameters $\tau$ that label the transformation stages of either a super-complex or an ultra-complex system, thus keeping track of the switches that occur between dynamic laws in highly complex dynamic systems with variable topology. Therefore, in the latter two cases, highly complex systems are in fact represented, respectively, by functor categories and supercategories of diagrams because categorical diagrams can be defined as functors. An important class of the simpler dynamic systems can be represented by algebraic categories; an example of such class of simple dynamic systems is that endowed with \emph{monadic dynamics} represented by the category of Eilenberg-Moore algebras.

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