categories of quantum LM-logic algebras CategoricalQuantumLMAlgebraicLogic 2008-12-14 22:32:38 2008-12-16 12:35:48 Topic categorical quantum logics LM-logic algebras % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym,color,enumerate} \usepackage{xypic} \xyoption{curve} \usepackage[mathscr]{eucal} %the next gives two direction arrows at the top of a 2 x 2 matrix 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\newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \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}} \newcommand{\hdgb}{\boldsymbol{\rho}^\square} \newcommand{\hdg}{\rho^\square_2} \renewcommand{\leq}{{\leqslant}} \renewcommand{\geq}{{\geqslant}} \def\red{\textcolor{red}} \def\magenta{\textcolor{magenta}} \def\blue{\textcolor{blue}} \def\<{\langle} \def\>{\rangle} \subsection{Categories of quantum \L{}M$-logic algebras} \subsubsection{Quantum Logics (QL) and Logic Lattice Algebras (LA)} \emph{Quantum Fields, General Relativity and Symmetries.} As the experimental findings in high-energy physics--coupled with theoretical studies-- have revealed the presence of new fields and symmetries, there appeared the need in modern physics to develop systematic procedures for generalizing space--time and Quantum State Space (QSS) representations in order to reflect these new concepts. In the General Relativity (GR) formulation, the local structure of space--time, characterized by its various tensors (of energy--momentum, torsion, curvature, etc.), incorporates the gravitational fields surrounding various masses. In Einstein's own representation, the physical space--time of GR has the structure of a Riemannian $R^4$ space over large distances, although the detailed local structure of space--time -- as Einstein perceived it -- is likely to be significantly different. On the other hand, there is a growing consensus in theoretical physics that a valid theory of Quantum Gravity requires a much deeper understanding of the small(est)--scale structure of Quantum Space--Time (QST) than currently developed. In Einstein's GR theory and his subsequent attempts at developing a unified field theory (as in the space concept advocated by Leibnitz), space-time does \emph{not} have an \emph{independent existence} from objects, matter or fields, but is instead an entity generated by the \emph{continuous} transformations of fields. Hence, the continuous nature of space--time was adopted in GR and Einstein's subsequent field theoretical developments. Furthermore, the quantum, or `quantized', versions of space-time, QST, are operationally defined through local quantum measurements in general reference frames that are prescribed by GR theory. Such a definition is therefore subject to the postulates of both GR theory and the axioms of Local Quantum Physics. We must emphasize, however, that this is \emph{not} the usual definition of position and time observables in `standard' QM. The general reference frame positioning in QST is itself subject to the Heisenberg uncertainty principle, and therefore it acquires through quantum measurements, a certain `fuzziness' at the Planck scale which is intrinsic to all microphysical quantum systems. Such systems with fuzziness include \emph{spin networks} that change in time thus giving birth to \emph{spin foam}. \emph{Lattices and Von Neumann-Birkhoff (VNB) Quantum Logic: Definition and Some Logical Properties.} We commence here by giving the \emph{set-based definition of a Lattice}. An \emph{s--lattice} $\mathbf{L}$, or a `set-based' lattice, is defined as a \emph{partially ordered set} that has all binary products (defined by the $s$--lattice operation `` $\bigwedge$") and coproducts (defined by the $s$--lattice operation ``$ \bigvee$ "), with the "partial ordering" between two elements X and Y belonging to the $s$--lattice being written as ``$X \preceq Y$". The partial order defined by $\preceq$ holds in \textbf{L }as $X \preceq Y$ if and only if $X = X \bigwedge Y $ (or equivalently, $Y = X \bigvee Y $ Eq.(3.1)(p. 49 of Mac Lane and Moerdijk, 1992). \emph{Operational Quantum Logic (OQL) and \L{}ukasiewicz Quantum Logic (LQL)} As pointed out by von Neumann and Birkhoff (1930), a logical foundation of quantum mechanics consistent with quantum algebra is essential for both the completeness and mathematical validity of the theory. The development of Quantum Mechanics from its very beginnings both inspired and required the consideration of specialized logics compatible with a new theory of measurements for microphysical systems. Such a specialized logic was initially formulated by von Neumann and Birkhoff (1932) and called `Quantum Logic'. Subsequent research on Quantum Logics (Chang, 1958; Genoutti, 1968; Dalla Chiara, 1968, 2004) resulted in several approaches that involve several types of non-distributive lattice (algebra) for $n$--valued quantum logics. Thus, modifications of the \L ukasiewicz Logic Algebras that were introduced in the context of algebraic categories by Georgescu and Vraciu (1973), also recently reviewed and expanded by Georgescu (2006), can provide an appropriate framework for representing quantum systems, or-- in their unmodified form- for describing the activities of complex networks in categories of \L{}ukasiewicz Logic Algebras (Baianu, 1977). \emph{\L{}ukasiewicz-Moisil (LM) Quantum Logic (LQL) and Algebras.} With all truth 'nuances' or assertions of the type $<<$ \emph{system A } is excitable to the $i$-th level and system B is excitable to the $j$-th level $>>$ one can define a special type of lattice which is subject to the axioms introduced by Georgescu and Vraciu ( 1970) and that becomes a \emph{$n$-valued \L ukasiewicz-Moisil, or LM, algebra}. Further algebraic and logic details are provided in Georgescu (2006) and Baianu et al (2007b). In order to have the $n$-valued \L{}ukasiewicz Logic Algebra represent correctly the basic behaviour of quantum systems (i.e., as observed through measurements that involve a quantum system interactions with a measuring instrument --which is a macroscopic object), several of these axioms have to be significantly changed so that the resulting lattice becomes \emph{non-distributive} and also (possibly) \emph{non--associative} (Dalla Chiara, 2004), in addition to being \emph{non-commutative}. With an appropriately defined quantum logic of events one can proceed to define Hilbert, or `nuclear'/Frechet, spaces in order to be able to utilize the `standard' procedures of quantum theories. \subsubsection{Fundamental Concepts of Space, Time and Space-Times in Quantum Theory vs. General Relativity} A notable feature of current 21-st century physical thought involves a close examination of the validity of the classical model of space-time as a $4$--dimensional manifold equipped with a Lorentz metric. The expectation of the earlier approaches to quantum gravity (QG) was to cope with microscopic length scales where a traditional manifold structure (in the conventional sense) needs to be forsaken (for instance, at the Planck length $L_p = (\frac{G\hslash}{c^3})^{\frac{1}{2}} \approx 10^{-35}m$). Whereas Newton, Riemann, Einstein, Weyl, Hawking, Penrose, Weinberg and many other exceptionally creative theoreticians regarded physical space as represented by a \emph{continuum}, there is an increasing number of proponents for a \emph{discrete, `quantized'} structure of space--time, since space itself is considered as discrete on the Planck scale. Like most radical theories, the latter view carries its own set of problems. The biggest problem arises from the fact that any discrete, `point-set' (or discrete topology), view of physical space--time is not only in immediate conflict with Einstein's General Relativity representation of space--time as a \emph{continuous Riemann} space, but it also conflicts with the fundamental impossibility of carrying out quantum measurements that would localize precisely either quantum events or masses at `singular points' (in the sense of disconnected, or isolated , sharply defined, geometric points) in space--time. Since GR seems to break down at the Planck scale, \emph{space--time may no longer be describable by a smooth manifold structure} such as a Riemann metric tensor. While not neglecting the large scale classical model, one needs to propose a structure of `ideal observations' as manifest in a limit, in some sense, of `discrete', or at least separable, measurements, where in such a limit it also encompasses the classical event. Further details are given in our recent, related paper (Baianu et al, 2007b). \emph{Deterministic Time--reversible-- vs. Probabilistic Time--Irreversibility and its Laws- Unitary vs. General Transformations} A significant part of the scientific--philosophical work of Ilya Prigogine (see e.g. Prigogine, 1980) has been devoted to the dynamical meaning of \emph{irreversibility} expressed in terms of the second law of thermodynamics. For systems with strong enough instability of motion, the concept of phase space trajectories is no longer meaningful and the dynamical description has to be replaced by the notion of distribution functions on the phase space. The viewpoint is that quantum theory produces a more coherent type of motion than in the classical setting, and the quantum effects induce correlations between neighbouring classical trajectories in phase space (which can be compared with the Bohr--Sommerfeld postulate of the image of phase cells having area $\hslash$). Prigogine's idea (1980) is to associate a macroscopic entropy (or Lyapounov function) with a microscopic entropy operator $M$~. One also notes the possibility of `contingent universes' with this `probabilistic time' paradigm. We must mention here that the time operator $T$ represents the `internal time', whereas the usual, `secondary' time in quantum dynamics is regarded as an average over $T$ (AN-2.6). Given the internal time's ability to distinguish between between future and past, a self-consistent scheme may be summarized in the following diagram (Prigogine, 1980): \begin{equation} \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=5pc{& {\text{Observer}} \ar[r] & {\text{Dynamics}} \ar[d] \\ &{\text{Broken time symmetry}} \ar[u] & \text{Dissipative structures} \ar[l] } \end{equation} for which `irreversibility' occurs as the intermediary in the following sequence: $$ \text{Dynamics} \Longrightarrow \text{Irreversibility} \Longrightarrow \text{Dissipative structures} $$ Note however that certain quantum theorists, including Einstein, regarded the irreversibility of time as an `illusion'. Others-- operating with minimal representations in quantum logic for finite quantum systems-- go further still by denying that there is any need for real time to appear in the formulation of quantum theory. \emph{Fundamental Concepts of Algebraic Topology with Potential Application to Ontology Levels Theory and Space-Time Structures.} We shall consider briefly the potential impact of novel Algebraic Topology concepts, methods and results on the problems of defining and classifying rigorously Quantum space-times. With the advent of Quantum Groupoids--generalizing Quantum Groups, Quantum Algebra and Quantum Algebraic Topology, several fundamental concepts and new theorems of Algebraic Topology may also acquire an enhanced importance through their potential applications to current problems in theoretical and mathematical physics, such as those described in an available preprint (Baianu, Brown and Glazebrook, 2006), and also in several recent publications (Baianu et al 2007a,b; Brown et al 2007). Now, if quantum mechanics is to reject the notion of a continuum, then it must also reject the notion of the real line and the notion of a path. How then is one to construct a homotopy theory? One possibility is to take the route signalled by \v{C}ech, and which later developed in the hands of Borsuk into `Shape Theory' (see, Cordier and Porter, 1989). Thus a quite general space is studied by means of its approximation by open covers. A few fundamental concepts of Algebraic Topology and Category Theory are summarized here that have an extremely wide range of applicability to the higher complexity levels of reality as well as to the fundamental, quantum level(s). Technical details are omitted in this section in order to focus only on the ontologically-relevant aspects; full mathematical details are however also available in a recent paper by Brown et al (2007) that focuses on a mathematical/conceptual framework for a completely formal approach to categorical ontology and the theory of levels. \subsubsection{Local--to--Global (LG) Construction Principles consistent with Quantum `Axiomatics'.} A novel approach to QST construction in Algebraic/Axiomatic QFT involves the use of generalized fundamental theorems of algebraic topology from specialized, `globally well-behaved' topological spaces, to arbitrary ones (Baianu et al, 2007c). In this category, are the generalized, \emph{Higher Homotopy van Kampen theorems (HHvKT)} of Algebraic Topology with novel and unique non-Abelian applications. Such theorems greatly aid the calculation of higher homotopy of topological spaces. R. Brown and coworkers (1999, 2004a,b,c) generalized the van Kampen theorem, at first to fundamental groupoids on a set of base points (Brown,1967), and then, to higher dimensional algebras involving, for example, homotopy double groupoids and 2-categories (Brown, 2004a). The more sensitive \emph{algebraic invariant} of topological spaces seems to be, however, captured only by \emph{cohomology} theory through an algebraic \emph{ring} structure that is not accessible either in homology theory, or in the existing homotopy theory. Thus, two arbitrary topological spaces that have isomorphic homology groups may not have isomorphic cohomological ring structures, and may also not be homeomorphic, even if they are of the same homotopy type. Furthermore, several \emph {non-Abelian} results in algebraic topology could only be derived from the Generalized van Kampen Theorem (\emph{viz}. Brown, 2004a), so that one may find links of such results to the expected \emph{`non-commutative} geometrical' structure of quantized space--time (Connes, 1994). In this context, the important algebraic--topological concept of a \emph{Fundamental Homotopy Groupoid (FHG) is applied to a Quantum Topological Space (QTS)} as a ``partial classifier" of the \emph{invariant} topological properties of quantum spaces of \emph{any} dimension; quantum topological spaces are then linked together in a \emph{crossed complex over a quantum groupoid} (Baianu, Brown and Glazebrook, 2006), thus suggesting the construction of global topological structures from local ones with well-defined quantum homotopy groupoids. The latter theme is then further pursued through defining locally topological groupoids that can be globally characterized by applying the Globalization Theorem, which involves the \emph{unique} construction of the Holonomy Groupoid. We are considering in a separate publication(Baianu et al 2007c) how such concepts might be applied in the context of Algebraic or Axiomatic Quantum Field Theory (AQFT) to provide a local-to-global construction of Quantum space-times which would still be valid in the presence of intense gravitational fields without generating singularities as in GR. The result of such a construction is a \emph{Quantum Holonomy Groupoid}, (QHG) which is unique up to an isomorphism. \subsection{Basic Structure of Categorical Ontology and the Theory of Levels. Emergence of Higher Levels and Their Sublevels} Here, we are in harmony with the theme and approach of the ontological theory of levels of reality (Poli, 1998, 2001) by considering a categorical, formal framework for Ontology and Poli's recent developments of the theory of levels. Thus our approach involves the mathematical techniques of category theory which afford describing the characteristics and binding of levels, besides representations of the links with other theories. Whereas Hartmann (1952) `stratified' levels in terms of four frameworks: physical, `organic'/biological, mental and `spiritual', we shall restrict mainly to the first three. The categorical techniques which we introduce provide a means of describing levels in both a linear and interwoven fashion thus leading to the necessary bill of fare: emergence, complexity and open non-equilibrium/irreversible systems. Furthermore, as shown by Baianu and Poli (2007), an effective approach to Philosophical Ontology is concerned with universal items assembled in categories of objects and relations, transformations and/or processes in general. Thus, Categorical Ontology is fundamentally dependent upon both space and time considerations. Basic concepts of Categorical Ontology are introduced in this section, whereas formal definitions were reported by Brown, Glazebrook and Baianu (2007). A dynamic classification of systems is also proposed for different levels of organization, beginning with the physical levels (including the fundamental quantum level) and continuing in an increasing order of complexity to the chemical/molecular levels, and then higher, towards the biological, psychological, societal and environmental levels. Indeed, it is in keeping with the basic tenet in the theory of levels that \emph{``there is a two-way interaction between social and mental systems that impinges upon the material realm for which the latter is the bearer of both"} (Poli, 2001). The evolution in our universe is thus seen to proceed from the level of `elementary' quantum `wave--particles', their interactions via quantized fields (photons, bosons, gluons, etc.), also including the quantum gravitation level, toward aggregates or categories of increasing complexity. In this sense, the classical macroscopic systems are defined as `simple' dynamical systems, computable recursively as numerical solutions of mathematical systems of either ordinary or partial differential equations. Underlying such mathematical systems is always the Boolean, or cryssippian, logic, namely, the logic of sets, Venn diagrams, digital computers and perhaps automatic reflex movements/motor actions of animals. The simple dynamical systems are always recursively computable (see for example, Suppes, 1995--2007), and in a certain specific sense, both degenerate and \emph{non-generic},consequently also \emph{structurally unstable} to small perturbations. The next higher order of systems is then exemplified by `systems with chaotic dynamics' that are conventionally called `complex' by physicists and computer scientists/modellers even though such physical, dynamical systems are still completely deterministic. It can be formally proven that such systems are \emph{recursively non-computable} (see for example, Baianu, 1987 for a 2-page, rigorous mathematical proof and relevant references), and therefore they cannot be completely and correctly simulated by digital computers, even though some are often expressed mathematically in terms of iterated maps or algorithmic-style formulas. In Section 5 we proceed to introduce the next higher level systems above the chaotic ones, which we shall call \emph{Super--Complex, Biological systems} (SCBS, or `organisms'), followed at still higher levels by the \emph{ultra-complex `systems'} (UCS) of the human mind and human societies that will be discussed in the last two sections. With an increasing level of complexity generated through billions of years of evolution in the beginning, followed by millions of years for the ascent of man, and perhaps ~10,000 more years for human societies and their civilizations, there is an increasing degree of \emph{genericity} for the dynamic states of the evolving systems (Thom, 1980; Rosen, 2001). The evolution to the next higher order of complexity- the ultra-complex `system' of processes--the human mind--may have become possible, and indeed accelerated, only through human societal interactions and effective, elaborate/rational and symbolic communication through speech (rather than screech-- as in the case of chimpanzees, gorillas). An effective Categorical Ontology requires, or generates--in the constructive sense--a `\emph{structure}' rather than a discrete set of items. The classification process itself generates collections of items, as well as a \emph{hierarchy of higher-level} \emph{`items'} of items, thus facing perhaps certain possible antimonies if such collections were to be just sets that are subject to the Axiom of Choice and problems arising from the set membership concept at different levels. The categorical viewpoint as emphasized by Lawvere, etc., is that the key structure is that of \emph{morphisms}, seen, for example, as abstract relations, mappings, functions, connections, interactions, transformations, etc. Therefore, in this section we shall consider both the Categorical viewpoint in the Ontology of Space and Time in complex/super-complex systems, as well as the fundamental structure of Categorical Ontology, as for example in the Ontological Theory of Levels (Poli, 2001; 2006a,b) which will be discussed briefly in the next section. \subsection{Theories: Axioms, Principles, Postulates and Laws.} The Greeks devised \emph{the axiomatic method}, but thought of it in a different manner to that we do today. One can imagine that the way Euclid's Geometry evolved was simply through the delivering of a course covering the established facts of the time. In delivering such a course, it is natural to formalize the starting points, and so arranging a sensible structure. These starting points came to be called \emph{postulates, definitions and axioms}, and they were thought to deal with real, or even ideal, objects, named points, lines, distance and so on. The modern view, initiated by the discovery of non Euclidean geometry, is that the words points, lines, etc. should be taken as undefined terms, and that axioms give the \emph{relations} between these. This allows the axioms to apply to many other instances, and has led to the power of modern geometry and algebra. Clarifying the meaning to be ascribed to `concept', `percept', `thought', `emotion', etc., and above all the \emph{relations} between these words, is clearly a fundamental but time--consuming step. Although relations--in their turn--can be, and were, defined in terms of sets, their axiomatic/categorical introduction greatly expands their range of applicability well beyond that of set-relations. Ultimately, one deals with \emph{relations among relations} and relations of higher order. The more rigorous scientific theories, including those founded in Logics and Mathematics, proceed at a fundamental level from axioms and principles, followed in the case of `natural sciences' by laws of nature that are valid in specific contexts or well-defined situations. Whereas the hierarchical theory of levels provides a powerful, systemic approach through categorical ontology, the foundation of science involves \emph{universal} models and theories pertaining to different levels of reality. Such theories are based on axioms, principles, postulates and laws operating on distinct levels of reality with a specific degree of complexity. Because of such distinctions, inter-level principles or laws are rare and over-simplified principles abound. As relevant examples, consider the Chemical/ Biochemical Thermodynamics, Physical Biochemistry and Molecular Biology fields which have developed a rich structure of specific-level laws and principles, however, without `breaking through' to the higher, emergent/integrative level of organismic biology. This does not detract of course from their usefulness, it simply renders them incomplete as theories of biological reality. With the possible exceptions of Evolution and Genetic Principles or Laws, Biology has until recently lacked other universal principles for highly complex dynamics in organisms, populations and species, as it will be shown in the following sections. One can therefore consider Biology to be at an almost `pre--Newtonian' stage by comparison with either Physics or Chemistry. Whereas axioms are rarely invoked in the natural sciences perhaps because of their abstract and exacting attributes, (as well as their coming into existence through elaborate processes of repeated abstraction and refinement), postulates are `obvious assumptions' of extreme generality that do not require proof but just like axioms are accepted on the basis of their very numerous, valid consequences. Principles and laws, even though quite strict, may not apply under certain exceptional, or `singular' situations. Natural laws are applicable to well-defined zones or levels of reality, and are thus less general, or universal, than principles. Unlike physical laws that are often expressed through mathematical equations, principles are instead often explained in words, and tend to have the most general form attainable/acceptable in an established theory. It is interesting to note that in Greek, and later Roman antiquity, both philosophers and orators did link philosophy and logic; moreover, in medieval time, first Francis Bacon, then Newton opted for quite precise formulations of ``natural philosophy" and a logical approach to `objective' reality. In Newton's approach, the logical and precise formulation of such ``natural principles" demanded the development of mathematical concepts suitable for the exact determination and quantification of the rate of a change in the ``state of motion" of any mechanical body, or system. Later philosophical developments have strayed from such precise formulations and, indeed, mathematical developments seem to have lost their appeal in `natural philosophy'. On the other hand, it would seem natural to expect that theories aimed at different ontological levels of reality should have different principles. Furthermore, one may ontologically, address the question of why such distinct levels of reality originated in the first place, and then developed, or emerged, both in space and time. Without reverting to any form of Newtonian or quantum-mechanical determinism, we are also pointing out in this essay the need for developing precise but nevertheless `flexible' concepts and novel mathematical representations suitable for understanding the emergence of the higher complexity levels of reality. It is also in this context that the `local-to-global' model approach becomes relevant, as in the case of generalized van Kampen theorems (see the Brown, Glazebrook and Baianu (2007) paper for a concise presentation of the van Kampen generalized theorems). Interestingly, the founder of Relational Biology, Nicolas Rashevsky (1968) proposed that physical laws and principles can be expressed in terms of \emph{mathematical functions}, or mappings, and are thus being predominantly expressed in a \emph{numerical} form, whereas the laws and principles of biological organisms and societies need take a more general form in terms of quite general, or abstract--mathematical and logical relations which cannot always be expressed numerically; the latter are often qualitative, whereas the former are predominantly quantitative. According to his suggested criterion, string theories may not be characteristic of the physical domain as they involve many qualitative relations and features. In this respect, one may also suggest that modern, Abstract Art, in its various forms-- if considered as a distinct class of representations--has moved ahead of modern philosophy to attempt universal representations of reality in a precise but flexible manner, thus appealing to both reason and emotions combined. It will be therefore worthwhile considering the structure of scientific theories and how it could be improved to enable the development of emergence principles for various complexity levels, including those of the \emph{inter(active)-level} types. The prejudice prevailing towards `pure', i.e. unmixed, levels of reality, and its detrimental effects on the development of Life sciences, Psychology, Sociology and Environmental sciences will be further discussed in subsequent sections. Then, alternatives and novel, possible solutions are presented in subsequent sections and the closing subsection of Brown et al (2007).\\ \emph{Towards Biological Postulates and Principles.} Often, Rashevsky considered in his Relational Biology papers, and indeed made comparisons, between established physical theories and principles. He was searching for new, more general relations in Biology and Sociology that were also compatible with the former. Furthermore, Rashevsky also proposed two biological principles that add to Darwin's natural selection of species and the `survival of the fittest principle', \emph{the emergent relational structure thus defining adaptive organisms}: \textbf{1. The Principle of Optimal Design},\\ and \textbf{2. The Principle of Relational Invariance} (phrased by Rashevsky as \emph{``Biological Epimorphism"}). In essence, the `Principle of Optimal Design' defines the `fittest' organism which survives in the natural selection process of competition between species, in terms of an extremal criterion, similar to that of Maupertuis; the optimally `designed' organism is that which acquires maximum functionality essential to survival of the successful species at the lowest `cost' possible. The `costs' are defined in the context of the environmental niche in terms of material, energy, genetic and organismic processes required to produce/entail the pre-requisite biological function(s) and their supporting anatomical structure(s) needed for competitive survival in the selected niche. Further details were presented by Robert Rosen in his short but significant book on optimality (1970). The `Principle of Biological Epimorphism' on the other hand states that the highly specialized biological functions of higher organisms can be mapped (through an epimorphism) onto those of the simpler organisms, and ultimately onto those of a (hypothetical) primordial organism (which was assumed to be unique up to an isomorphism or \emph{selection-equivalence}). The latter proposition, as formulated by Rashevsky, is more akin to a postulate than a principle. However, it was then generalized and re-stated in the form of the existence of a \emph{limit} in the category of living organisms and their functional genetic networks ($\textbf{GN}^i$), as a directed family of objects, $\textbf{GN}^i(-t)$ projected backwards in time (Baianu and Marinescu, 1968), or subsequently as a super-limit (Baianu, 1970 to 1987; Baianu, Brown, Georgescu and Glazebrook, 2006); then, it was re-phrased as the \emph{Postulate of Relational Invariance}, represented by a \emph{colimit} with the arrow of time pointing forward (Baianu, Brown, Georgescu and Glazebrook, 2006). Somewhat similarly, a dual principle and colimit construction was invoked for the ontogenetic development of organisms (Baianu, 1970), and also for populations evolving forward in time; this was subsequently applied to biological evolution although on a much longer time scale --that of evolution-- also with the arrow of time pointing towards the future in a representation operating through Memory Evolutive Systems (MES) by A. Ehresmann and Vanbremeersch (2006). \subsection{The Object-Based Approach \emph{vs} Process-Based, \emph{Dynamic Ontology}.} In classifications, such as those developed over time in Biology for organisms, or in Chemistry for chemical elements, the \emph{objects} are the basic items being classified even if the `ultimate' goal may be, for example, either evolutionary or mechanistic studies. Rutherford's comment is pertinent in this context: \emph{``There are two major types of science: physics or stamp collecting." } An ontology based strictly on object classification may have little to offer from the point of view of its cognitive content. It is often thought or taken for granted that the \emph{object-oriented} approach can be readily converted into a process-based one. It would seem, however, that the answer to this question depends critically on the ontological level selected. For example, at the quantum level, \emph{object and process become inter-mingled}. Either comparing or moving between levels, requires ultimately a \emph{process-based} approach, especially in Categorical Ontology where relations and inter-process connections are essential to developing any valid theory. At the fundamental level of `elementary particle physics' however the answer to this question of process-vs. object becomes quite difficult as a result of the `blurring' between the particle and the wave concepts. Thus, it is well-known that any `elementary quantum object' is considered by all accepted versions of quantum theory not just as a `particle' or just a `wave' but both: the quantum `object' is \emph{both} wave and particle, \emph{at the same-time}, a proposition accepted since the time when it was proposed by de Broglie. At the quantum microscopic level, the object and process are inter-mingled, they are no longer separate items. Therefore, in the quantum view the `object-particle' and the dynamic process-`wave' are united into a single dynamic entity or item, called \emph{the wave-particle quantum}, which strangely enough is \emph{neither discrete nor continuous}, but both at the same time, thus `refusing' intrinsically to be an item consistent with Boolean logic. Ontologically, the quantum level is a fundamentally important starting point which needs to be taken into account by any theory of levels that aims at completeness. Such completeness may not be attainable, however, simply because an `extension' of G\"odel's theorem may hold here also. The fundamental quantum level is generally accepted to be dynamically, or intrinsically \emph{non-commutative}, in the sense of the\emph{ non-commutative quantum logic} and also in the sense of \emph{non-commuting quantum operators} for the essential quantum observables such as position and momentum. Therefore, any comprehensive theory of levels, in the sense of incorporating the quantum level, is thus --\emph{mutatis mutandis}-- \emph {non-Abelian}. Furthermore, as the non-Abelian case is the more general one, from a strictly formal viewpoint, a non-Abelian Categorical Ontology is the preferred choice. A paradigm-shift towards a \emph{non-Abelian Categorical Ontology} has already started (Brown et al, 2007: \emph{`Non-Abelian Algebraic Topology'}; Baianu, Brown and Glazebrook, 2006: NA-QAT; Baianu et al 2007a,b,c).\\ \subsection{Towards a Formal Theory of Levels.} The first subsection here will present the fundamentals of the ontological theory of levels together with its further development in terms of mathematical categories, functors and natural transformations, as well as the necessary non-commutative generalizations of Abelian categorical concepts to non-Abelian formal systems and theories. \subsubsection{Fundamentals of Poli's Theory of Levels.} The ontological theory of levels (Poli, 2001, 2006a,b; 2007) considers a hierarchy of \emph{items} structured on different levels of existence with the higher levels \emph{emerging} from the lower, but usually \emph{not} reducible to the latter, as claimed by widespread reductionism. This approach draws from previous work by Hartmann (1935,1952) but also modifies and expands considerably both its vision and range of possibilities. Thus, Poli (1998, 2001a, 2006a,b; 2007) considers four realms or \emph{levels} of reality: Material-inanimate/Physico-chemical, Material-living/Biological, Psychological and Social. We harmonize this theme by considering categorical models of complex systems in terms of an evolutionary dynamic viewpoint using the mathematical methods of category theory which afford describing the characteristics and binding of levels, besides the links with other theories which, \emph{a priori}, are essential requirements. The categorical techniques which form an integral part of the discussion provide a means of describing a hierarchy of levels in both a linear and interwoven, or \emph{entangled}, fashion, thus leading to the necessary bill of fare: emergence, higher complexity and open, non-equilibrium/irreversible systems. We further stress that the categorical methodology intended is \emph{intrinsically `higher dimensional'} and can thus account for `processes between processes...' within, or between, the levels--and sub-levels-- in question. Whereas a strictly Boolean classification of levels allows only for the occurrence of \emph{discrete} ontological levels, and also does not readily accommodate either \emph{contingent} or \emph{stochastic sub-levels}, the LM-logic algebra is readily extended to \emph{continuous}, \emph{contingent }or even \emph{fuzzy} (Baianu and Marinescu, 1968) sub-levels, or levels of reality (cf. Georgescu, 2006; Baianu, 1977, 1987; Baianu, Brown, Georgescu and Glazebrook, 2006). Clearly, a Non-Abelian Ontology of Levels would require the inclusion of either Q- or LM- logics algebraic categories because it begins at the fundamental quantum level --where Q-logic reigns-- and `rises' to the emergent ultra-complex level(s) with `all' of its possible sub-levels represented by certain LM-logics. Poli (2006a) has stressed a need for understanding \emph{causal and spatiotemporal }phenomena formulated within a \emph{descriptive categorical context} for theoretical levels of reality. There are three main points to be taken into account: differing spatiotemporal regions necessitate different (levels of) causation, for some regions of reality analytic reductionism may be inadequate, and there is the need to develop a \emph{synthetic} methodology in order to compensate for the latter, although one notes (v. Rosen, 2001) that \emph{analysis and synthesis are not the exact inverse of each other}. Following Poli (2001), we consider a causal dependence on levels, somewhat apart from a categorical dependence. At the same time, we address the \emph{internal dynamics}, the \emph{temporal rhythm, or cycles}, and the subsequent unfolding of reality. The genera of corresponding concepts such as `processes', `groups', `essence', `stereotypes', and so on, can be simply referred to as \emph{`items}' which allow for the existence of many forms of causal connection (Poli, 2007). The implicit meaning is that the \emph{irreducible multiplicity} of such connections converges, or it is ontologically integrated within a \emph{unified synthesis}. Rejecting reductionism thus necessitates accounting for an \emph{irreducible multiplicity of ontological levels}, and possibly the ontological acceptance of many worlds also. In this regard, the Brentano hypothesis is that the class of physical phenomena and the class of psychological (or spiritual) phenomena are \emph{complementary}; in other words, physical categories were said to be `orthogonal' to psychological categories (Poli, 2006a,b). As befitting the situation, there are devised \emph{universal} categories of reality in its entirety, and also subcategories which apply to the respective sub-domains of reality. Following Poli (2001), the ontological procedures in question provide: \begin{itemize} \item coordination between categories (for instance, the interactions and parallels between biological and ecological reproduction as in Poli, 2001); \item modes of dependence between levels (for instance, how the co-evolution/interaction of social and mental realms depend and impinge upon the material); \item the categorical closure (or completeness) of levels. \end{itemize} Already we can underscore a significant component of this Topic that relates the ontology to geometry and topology; specifically, if a level is defined \emph{via} `iterates of local procedures' (viz. `items in iteration', Poli, 2001), then we have some handle on describing its intrinsic governing dynamics (with feedback ) and, to quote Poli (2001), to `restrict the \emph{multi-dynamic} frames to their linear fragments'. On each level of this ontological hierarchy there is a significant amount of connectivity through inter-dependence, interactions or general relations often giving rise to complex patterns that are not readily analyzed by partitioning or through stochastic methods as they are neither simple, nor are they random connections. But we claim that such complex patterns and processes have their logico-categorical representations quite apart from classical, Boolean mechanisms. This ontological situation gives rise to a wide variety of networks, graphs, and/or mathematical categories, all with different connectivity rules, different types of activities, and also \emph{a hierarchy of super-networks of networks of sub-networks}. Then, the important question arises what types of basic symmetry or patterns such super-networks of items can have, and also how do the effects of their sub-networks `percolate' through the various levels. From the categorical viewpoint, these are of two basic types: they are either \emph{commutative} or \emph{non-commutative}, where, at least at the quantum level, the latter takes precedence over the former, as we shall further discuss and explain in the following sections.\\ \emph{Proceeding from Lower to Higher Order Theories.} In accordance with replacing reductionism by appropriate complexity theories of the highly complex human mind and its supporting matter systems in the brain, one requires second order models consisting of a \emph{meta--model} or \emph{meta--theory}. A brief and only partial analogy as discussed in Atmanspacher and Jahn (2003) might be made with first--order engineering connecting hardware to software in AI systems; this partial analogy suffers, however, from severe, reductionist limitations. In a separate context, the expectation value of an observable defined in some limit $N \lra \infty$, which conceivably does not exist, in the second order viewpoint can be realized by studying the mean-value of the considered observation as changing in accord with functions up to finite $N$. In general it is erroneous to employ first-order experiments as an attempt to validate second-order models (a psychological stumbling block when it comes to \emph{``thinking about thinking''}, again, viz. the `mereological fallacy', Bennett and Hacker, 2003). In other words, whereas a level $(n-1)$-theory may be deducible from a level $n$-theory, \emph{the converse is not true}, in so far, for instance, that a theory of neuronal assemblies cannot be used as the sole basis for the explanation of a given cognitive process. In this regard, the categorical methods we propose for (ultra) complex systems are suitably geared for the `contraction principle' in going from level $n$ down to level $(n-1)$ and making the right predictions accordingly. For example, the `self' increases in complexity in confronting new challenges and implementing new tasks. But this categorical approach of access to level $n-1$ from level $n$ is a blueprint for studying complex processes that the usual `self' often dispenses with. Many individuals can admirably perform their secular duties, enjoy their leisure etc in society without any due regard to the concepts and functions of their corporal metabolism, neurophysiology, and cognitive mechanisms, etc., unless illness or some other disposition causes an alert to these functions. The situation for AI and `conscious' machines is even more pronounced. Chalmers (1996) points out the examples of Hofstadter (1979) -- it is not necessary to give a system access to its low-level components - and Winograd's program SHRDLU (1972) had no knowledge of the programming language in which it was written despite its capacity to assimilate the structure of a virtual world and make inferences about it. \subsubsection{Towards a Process-Based, Dynamic Ontology} \emph{From Object and Structure to Organismic Functions and Relations.} Although the essence of super-- and ultra-- complex systems is in the \emph{interactions, relations and dynamic transformations} that are ubiquitous in such higher--level ontology, surprisingly many a psychology, cognitive and an ontology approach begins with a very strong emphasis on \emph{objects} rather than relations. It would also seem that a basic `trick' of human consciousness is to pin a subjective sensation, perception and/or feeling on an internalized \emph{object}, or vice-versa to represent/internalize an object in the form of an internal symbol in the mind. The example often given is that of a human child's substituting a language symbol, or image for the \emph{mother `object'}, thus allowing `her permanent presence' in the child's consciousness. Clearly, however, a complete approach to ontology must also include \emph{relations and interconnections} between items, with a strong emphasis on \emph{dynamic processes, complexity} and \emph{functionality} of systems, which all require an emphasis on general relations, \emph{morphisms} and the \emph{categorical viewpoint} of ontology. \emph{Physico-chemical Structure--Function Relationships} Perhaps an adequate response to both physicalist reductionism and/or `pure' relationalism (as defined here in the previous sections) consists in considering the integration of a concrete categorical ontology approach which considers important experimentally well- studied examples of super-complex systems of defined physico-chemical structures with organizational--relational/ logical-abstract models that are expressed in terms of related function(s). Whereas such a combined approach does address the needs of-- and in fact it is essential to-- the experimental science of complex/super-complex systems, it is also considerably more difficult than either physicalist reductionism, \emph{abstract relationalism} or `rhetorical mathematics'. Moreover, because there are many alternative ways in which the physico-chemical structures can be combined within an organizational map or relational complex system, there is a \emph{multiplicity of `solutions'} or mathematical models that needs be investigated, and the latter are not computable with a digital computer in the case of complex/super-complex systems such as organisms (Rosen 1987). It is generally accepted at present that \emph{structure-functionality relationships are key to the understanding of super-complex systems} such as living cells and organisms. This classification problem of structure-functionality classes for various organisms and various complex models is therefore a difficult and yet unresolved one, even though several paths and categorical methods may lead to rapid progress in Categorical Ontology as discussed here in Section 3. The problem is further compounded by the presence of structural \emph{disorder} (in the physical structure sense) which leads to a \emph{multiplicity} of dynamical-physicochemical structures (or `configurations') of a biopolymer, be it a protein, enzyme, or nucleic acid in a living cell or organism that correspond, or `realize', just a single recognizable biological function (Baianu, 1980b); this complicates the assignment of a `fuzzy' physico-chemical structure to a well-defined biological function unless extensive experimental data are available, as for example, those derived through computation from 2D-NMR spectroscopy data (W\"utrich, 1996), or neutron/X-ray scattering and related multi-nuclear NMR spectroscopy/relaxation data (as for example in Chapters 2 to 9 in Baianu et al., 1995). It remains to be seen if this approach can also be carried \emph{in vivo} in specially favorable cases. Detailed considerations of the ubiquitous, partial disorder effects on the structure-functionality relationships were reported for the first time by Baianu (1980b). Specific aspects were also recently discussed by W\"utrich (1996) on the basis of 2D-FT NMR analysis. \subsection{The Categorical Ontology of Levels} In the following subsections we shall outline a Categorical Framework for the Ontological Theory of Levels. \subsubsection{Categorical Representations of the Ontological Theory of Levels: A Paradigm Shift from Abelian Categories to Non-Abelian Theories.} General system analysis seems to require formulating ontology by means of categorical concepts (Poli, 2007, TAO-1; Baianu and Poli, 2007). Furthermore, category theory appears as a natural framework for any general theory of transformations or dynamic processes, just as group theory provides the appropriate framework for classical dynamics and quantum systems with a finite number of degrees of freedom. Therefore, we shall adopt here a categorical approach as the starting point, meaning that we are looking for \emph{``what is universal"} (in some domain, or in general), and that for simple systems this involves \emph{commutative} modelling diagrams and structures (as, for example, in Figure 1 of Rosen, 1987). Note that this ontological use of the word \emph{`universal'} is quite distinct from the mathematical use of \emph{`universal property'}, which means that a property of a construction on particular objects is defined by its relation to \emph{all} other objects (i.e., it is a \emph{global} attribute), usually through constructing a morphism, since this is the only way, in an \emph{abstract} category, for objects to be related. With the first (ontological) meaning, the most universal feature of reality is that it is \emph{temporal}, i.e. it changes, it is subject to countless transformations, movements and alterations. In this select case of \emph{universal temporality}, it seems that the two different meanings can be brought into superposition through appropriate formalization. Furthermore, \emph{concrete} categories may also allow for the representation of ontological `universal items' as in certain previous applications to \emph{cat-neurons}-- categories of neural networks (Baianu, 1972; Ehresmann and Vanbremeersch, 2006). For general categories, however, each object is a kind of a Skinnerian black box, whose only exposure is through input and output, i.e. the object is given by its \emph{connectivity} through various morphisms, to other objects. For example, the opposite of the category of sets has objects but these have \emph{no structure} from the categorical viewpoint. Other types of category are important as expressing useful relationships on structures, for example \emph{lextensive} categories, which have been used to express a general van Kampen theorem by Brown and Janelidze (1997). This concrete categorical approach seems also to provide an elegant formalization that matches the ontological theory of levels briefly described above. The major restriction--as well as for some, attraction-- of the 3-level categorical construction outlined above seems to be its built-in \emph{commutativity} (see also Section 3.2 for further details). Note also how level-2 arrows become level-`3 objects' in the meta--category, or level-`3' category, of functors and natural transformations. This construction has already been considered to be suitable for representing dynamic processes in a \emph{generalized Quantum Field Theory }(G-QFT). The presence of mathematical structures is just as important for highly complex systems, such as organisms, whose organizational structure--in this mathematical and biological function/physiological sense--may be superficially apparent but difficult to relate unequivocally to anatomical, biochemical or molecular `structures'. Thus, abstract mathematical structures are developed to define \emph{relationships}, to deduce and calculate, to exploit and define analogies, since \emph{analogies are between relations} between things rather than between things themselves. As \emph{structures} and \emph{relations} are present at the very core of mathematical developments (Ehresmann,1965; 1967), the theories of categories and toposes (topoi) distinguish at least two fundamental types of items: \emph{objects} and \emph{arrows} (also called suggestively \emph{`morphisms'}). Thus, first-level arrows may represent mappings, relations, interactions, dynamic transformations, and so on, whereas categorical objects are usually endowed with a selected type of structure only in \emph{`concrete' categories of `sets with structure'}. Note, however, that simple sets have only the \emph{`discrete topology structure}', consisting of just discrete elements, or points (sometimes called \emph{`set dust'} by its critics). A description of a new structure is in some sense a development of part of a new `language'. The notion of \emph{structure} is also related to the notion of \emph{analogy}. One must note in the latter case above the use of a very different meaning of the word `structure', quite distinct from that of the organizational/physiological and mathematical structure introduced at the beginning of this section. Even though concrete, molecular or anatomical `structures' could also be defined with the help of `concrete sets with structure', the physical structures representing `anatomy' are very different from those representing physiological-functional/organizational structures. It is one of the triumphs of the mathematical theory of categories in the 20th century to make progress in \emph{unifying} mathematics through the finding of \emph{analogies} between various behavior of structures across different areas of mathematics. This theme is further elaborated in the article by Brown and Porter (2002) which argue that many analogies in mathematics, and in many other areas, are \emph{not} between objects themselves but \emph{between the relations} between objects. Here, we mention as an example, only the categorical notion of a \emph{pushout}, which we shall use later in discussing the higher homotopy, generalized van Kampen theorems. A \emph{pushout} has the same definition in different categories even though the construction of pushouts in these categories may be widely different. Thus, focusing on the \emph{constructions} rather than on the \emph{universal properties} may lead to a failure to see the analogies. Super-pushouts, on the other hand, were reported to be involved in multi-stability and metamorphoses of living organisms (Baianu, 1970). Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures. For example, we are also seeking \emph{non-Abelian} methods for higher dimensional local-to-global problems in homotopy theory (Brown et al. 2007). In reference to the above discussion, one of the major goals of category theory is to see how the properties of a particular mathematical structure, say $S$, are reflected in the properties of the category $\mathsf{Cat}(S)$ of all such structures and of morphisms between them. Thus the first step in category theory is that a definition of a structure should come with a definition of a morphism of such structures. Usually, but not always, such a definition is obvious. The next step is to compare structures. This might be obtained by means of a \emph{functor} $A: \mathsf{Cat}(S) \lra \mathsf{Cat}(T)$. Finally, we want to compare such functors $A,B: \mathsf{Cat}(S) \lra \mathsf{Cat}(T)$. This is done by means of a natural transformation $\eta: A \Rightarrow B$. Here $\eta$ assigns to each object $X$ of $ \mathsf{Cat}(S)$ a morphism $\eta(X): A(X) \lra B(X)$ satisfying a commutativity condition for any morphism $a: X \lra Y$. In fact we can say that $\eta$ assigns to each morphism $a$ of $\mathsf{Cat}(S)$ a commutative square of morphisms in $\mathsf{Cat}(T)$ (as shown in Diagram 13.2 in the Brown, Glazebrook and Baianu (2007).). This notion of \emph{natural transformation} is at the heart of category theory. As Eilenberg-Mac Lane write: ``\emph{to define natural transformations one needs a definition of functor, and to define the latter one needs a definition of category".} From the point of view of mathematical modelling, the mathematical theory of categories models the dynamical nature of reality by representing temporal changes through either \emph{variable} categories or through \emph{toposes}. \subsubsection{A Hierarchical, Formal Theory of Levels. Commutative and Non-Commutative Structures: Abelian Category Theory vs. Non-Abelian Theories.} One could formalize-for example as outlined in Baianu and Poli (2008, in this volume)--the hierarchy of multiple-level relations and structures that are present in biological, environmental and social systems in terms of the mathematical Theory of Categories, Functors and Natural Transformations (TC-FNT, see Brown, Glazebrook and Baianu (2007).). On the first level of such a hierarchy are the links between the system components represented as \emph{`morphisms'} of a structured category which are subject to several axioms/restrictions of Category Theory, such as \emph{commutativity} and associativity conditions for morphisms, functors and natural transformations. Among such mathematical structures, \emph{Abelian} categories have particularly interesting applications to rings and modules (Popescu, 1973; Gabriel, 1962) in which commutative diagrams are essential. Commutative diagrams are also being widely used in Algebraic Topology (Brown, 2005; May, 1999). Their applications in computer science also abound. Then, on the second level of the hierarchy one considers \emph{`functors'}, or links, between such first level categories, that compare categories without 'looking inside' their objects/ On the third level, one compares, or links, functors using \emph{`natural transformations'} in a level-`3' category (meta-category) of functors and natural transformations. At this level, natural transformations not only compare functors but also look inside the first level objects (system components) thus 'closing' the structure and establishing `the universal links' between items as an integration of both first and second level links between items. The advantages of this constructive approach in the mathematical theory of categories, functors and natural transformations have been recognized since the beginnings of this mathematical theory in the seminal paper of Mac Lane and Eilenberg (1945). Note, however, that \emph{in general categories the objects have no `inside', even though they may do so for example in the case of `concrete' categories or in topoi}. \emph{Symmetry, Commutativity and Abelian Structures.} The hierarchy constructed above, up to level 3, can be further extended to higher, $n$-levels, always in a consistent, natural manner, that is using commutative diagrams. Let us see therefore a few simple examples or specific instances of commutative properties. The type of global, natural hierarchy of items inspired by the mathematical TC-FNT has a kind of \emph{internal symmetry} because at all levels, the link compositions are \emph{natural}, that is, if $f: x \lra y$ and $g: y \lra z \Longrightarrow h: x \lra z$, then the composition of morphism $g$ with $f$ is given by another unique morphism $h = g \circ f$. This general property involving the equality of such link composition chains or diagrams comprising any number of sequential links between the same beginning and ending objects is called \emph{commutativity} (see for example Samuel and Zarisky, 1957), and is often expressed as a \emph{naturality condition for diagrams}. This key mathematical property also includes the mirror-like symmetry $x\star y = y\star x$; when $x$ and $y$ are operators and the symbol '$\star$' represents the operator multiplication. Then, the equality of $x\star y$ with $y\star x$ defines the statement that "the $x$ and $y$ operators \emph{commute}"; in physical terms, this translates into a sharing of the same set of eigenvalues by the two commuting operators, thus leading to `equivalent' numerical results {i.e., up to a multiplication constant); furthermore, the observations X and Y corresponding, respectively, to these two operators would yield the same result if X is performed before Y in time, or if Y is performed first followed by X. This property, when present, is very convenient for both mathematical and physical applications (such as those encountered in quantum mechanics). When commutativity is global in a structure, as in an Abelian (or commutative) group, commutative groupoid, commutative ring, etc., such a structure that is commutative throughout is usually called \textbf{\emph{Abelian}}. However, in the case of category theory, this concept of Abelian structure has been extended to a special class of categories that have meta-properties formally similar to those of the category of commutative groups, \emph{Ab}-\textbf{G}; the necessary and sufficient conditions for such `Abelianness' of categories other than that of Abelian groups were expressed as three axioms \textbf{Ab1} to \textbf{Ab3 } and their duals (Freyd, 1964; see also the details in Baianu et al 2007b and Brown et al 2007). A first step towards re-gaining something like the `global commutativity' of an Abelian group is to require that all classes of morphisms [A,B] or Hom(A,B) have the structure of commutative groups; subject to a few other general conditions such categories are called \textbf{additive}. Then, some kind of global commutativity is assured for all morphisms of \emph{additive } categories. However, further conditions are needed to make additive categories `Abelian', and additional properties were also posited for Abelian categories in order to extend the applications of Abelian category theory to other fields of modern mathematics (Grothendieck, 1957; Grothendieck and Dieudon\'{e} 1960; Oberst 1969; Popescu 1973.) A Homotopy theory was also formulated in Abelian categories (Kleisli, 1962). The equivalence of Abelian categories was reported by Roux, and important imbedding theorems were proved by Mitchell (1964) and by Lubkin (1960); a characterization of Abelian categories with generators and exact limits was presented by Gabriel and Popescu (1964). As one can see from both earlier and recent literature, Abelian categories have been studied in great detail, even though one cannot say that all their properties have been already found. However, not all quantum operators `commute', and not all categorical diagrams or mathematical structures are, or need be, commutative. \emph{Non-commutativity} may therefore appear as a result of `breaking' the `internal symmetry' represented by commutativity. As a physical analogy, this might be considered a kind of \emph{`symmetry breaking'} which is thought to be responsible for our expanding Universe and CPT violation, as well as many other physical phenomena such as phase transitions and superconductivity (Weinberg, 2003). \begin{thebibliography}{9} \bibitem{ICB-RO2k8} Baianu, I.C. and R. Poli. (2008), From Simple to Complex and Ultra-Complex Systems: A Paradigm Shift Towards Non-Abelian Systems Dynamics., In: {\em Theory and Applications of Ontology}, vol. \textbf{1}, R. Poli, Ed., Springer: Berlin. \bibitem{BGB2k7b} Brown, R., Glazebrook, J. F. and I.C. Baianu.(2007), A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly Complex Structures and Dynamics., \emph{Axiomathes} (17): 321--379. \bibitem{BBGG1} Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook.(2006), Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L{}ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks, \emph{Axiomathes}, \textbf{16} Nos. 1--2: 65--122. \bibitem{Bggb4} Baianu, I.C., R. Brown and J. F. Glazebrook.(2007), A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, {\em Axiomathes}, \textbf{17}: 169-225. \bibitem{Poli1} Poli, R. (1998), Levels,\emph{Axiomathes}, \textbf{9}, 1-2, pp. 197-211. \bibitem{Poli2} Poli, R. (2001a), The Basic Problem of the Theory of Levels of Reality, \emph{Axiomathes}, \textbf{12}, 3-4, pp. 261-283. \bibitem{Poli3} Poli, R. (2001b), Alwis. {\em Ontology for Knowledge Engineers}. PhD Thesis, Univ. of Utrecht. \end{thebibliography}