HighlyComplexSystems 2009-05-02 14:44:19 2026-09-07 03:10:25 Definition section edits ultra-complex systems % almost certainly you want these \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsthm} % define commands here % Commutative/categorical diagrams \usepackage[all]{xy} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] \theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} % Frequently used operators \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Cl}{\operatorname{Cl}} \newcommand{\Co}{\operatorname{Co}} \newcommand{\DES}{\operatorname{DES}} \newcommand{\Diff}{\operatorname{Diff}} \newcommand{\Dom}{\operatorname{Dom}} \newcommand{\Hol}{\operatorname{Hol}} \newcommand{\Mon}{\operatorname{Mon}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ker}{\operatorname{Ker}} \newcommand{\Ind}{\operatorname{Ind}} \newcommand{\IM}{\operatorname{Im}} \newcommand{\Is}{\operatorname{Is}} \newcommand{\ID}{\operatorname{id}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sem}{\operatorname{Sem}} \newcommand{\St}{\operatorname{St}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\SU}{\operatorname{SU}} \newcommand{\Tor}{\operatorname{Tor}} \newcommand{\U}{\operatorname{U}} % Calligraphic symbols \newcommand{\A}{\mathcal{A}} \newcommand{\Ce}{\mathcal{C}} \newcommand{\D}{\mathcal{D}} \newcommand{\E}{\mathcal{E}} \newcommand{\F}{\mathcal{F}} \newcommand{\G}{\mathcal{G}} \newcommand{\Q}{\mathcal{Q}} \newcommand{\R}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cU}{\mathcal{U}} \newcommand{\W}{\mathcal{W}} % Blackboard bold symbols \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\grp}{\mathbb{G}} \newcommand{\dgrp}{\mathbb{D}} \newcommand{\desp}{\mathbb{D}^{\mathrm{es}}} \newcommand{\Geod}{\operatorname{Geod}} \newcommand{\geod}{\operatorname{geod}} \newcommand{\hgr}{\mathbb{H}} \newcommand{\mgr}{\mathbb{M}} \newcommand{\ob}{\operatorname{Ob}} \newcommand{\obg}{\operatorname{Ob}(\mathbb{G})} \newcommand{\obgp}{\operatorname{Ob}(\mathbb{G}')} \newcommand{\obh}{\operatorname{Ob}(\mathbb{H})} \newcommand{\Osmooth}{\Omega^{\infty}(X,*)} \newcommand{\ghomotop}{\rho_2^{\square}} \newcommand{\gcalp}{\mathbb{G}(\mathcal{P})} \newcommand{\rf}{R_{\mathcal{F}}} \newcommand{\glob}{\operatorname{glob}} \newcommand{\loc}{\operatorname{loc}} \newcommand{\TOP}{\operatorname{TOP}} \newcommand{\wti}[1]{\widetilde{#1}} \newcommand{\what}[1]{\widehat{#1}} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{\longrightarrow} \newcommand{\ra}{\rightarrow} \newcommand{\rat}{\rightarrowtail} \newcommand{\oset}[1]{\overset{#1}{\ra}} \newcommand{\osetl}[1]{\overset{#1}{\lra}} \newcommand{\hr}{\hookrightarrow} \section{Highly-complex systems} \subsubsection{Introduction} Modeling the emergence of the ultra-complex system of the human mind, based on the super-complex human organism, one needs to consider 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. One can represent by square categorical diagrams the emergence of ultra-complex dynamics from the super-complex dynamics of human organisms coupled \emph{via} social interactions in characteristic patterns represented by \PMlinkname{Rosetta biogroupoids}{RosettaGroupoids}, together with complex, albeit inanimate, systems with ``chaos,'' as discussed next. \subsection{Diagrams of simple and highly-complex systems} \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'': \[ \xymatrix@C=5pc@R=3pc{ \text{SUPER-COMPLEX} \ar[r]^{\text{Higher Dim.}} \ar[d]_{\Lambda} & U_{CS}=\text{ULTRA-COMPLEX} \ar[d]^{\text{onto}} \\ \text{COMPLEX} & \text{SIMPLE} \ar[l]^{\text{Generic Map}} } \] \end{definition} \subsection{Remarks} Note 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 or organismic level and the ultra-complex psychological or neural-network-dynamic level, in comparison with the low dimensions of either simple physical or classical systems or complex chaotic dynamic systems. An \emph{ultra-complex system} represents the human mind and consciousness from the standpoint of a 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.