A CATEGORY THEORY AND HIGHER DIMENSIONAL ALGEBRA APPROACH TO COMPLEX SYSTEMS BIOLOGY, META-SYSTEMS AND ONTOLOGICAL THEORY OF LEVELS:

A CATEGORY THEORY AND HIGHER DIMENSIONAL ALGEBRA APPROACH TO COMPLEX SYSTEMS BIOLOGY, META-SYSTEMS AND ONTOLOGICAL THEORY OF LEVELS:EMERGENCE OF LIFE, SOCIETY, HUMAN CONSCIOUSNESS AND ARTIFICIAL INTELLIGENCE

I. C. Baianu, James F. Glazebrook and Ronald Brown

Abstract.

An attempt is made from the viewpoint of the recent theory of ontological levels [2],[40],[137],[206]-[209]to understand the origins and emergence of life, the dynamics of the evolution of organisms and species, theascent of man and the co-emergence, as well as co-evolution of human consciousness within organised societies.The new concepts developed for understanding the emergence and evolution of life, as well as human consciousness,are in terms of globalisation of multiple, underlying processes into the meta-levels of their existence.Such concepts are also useful in computer aided ontology and computer science [1],[194],[197].In this monograph we present a novel approach to the problems raised by higher complexity in both nature and the human society, byconsidering the highest and most complex levels of objective existence as ontological meta-levels, such as those present in thecreative human minds and civilised, modern societies.Thus, a collection of sets may be a class, instead of a set [59],[176]-[177]; itmay also be called a ‘super-set’, or a meta-set; a ‘theorem’about theorems is a meta-theorem, and a ‘theory’ abouttheories is a ‘meta-theory’. In the same sense that astatement about propositions is a higher-level $\\langle proposition\\rangle$ rather than a simple proposition, a global process of subprocessesis a meta-process, and the emergence of higher levels ofreality via such meta-processes results in the objectiveexistence of ontological meta-levels. It is also attempted here to classify more precisely the levels of reality andspecies of organisms than it has been thus far reported. Theselected approach for our broad– but in-depth– study of thefundamental, relational structures and functions present in living,higher organisms and of the extremely complex processes andmeta-processes of the human mind combines new concepts from threerecently developed, related mathematical fields: Algebraic Topology,Category Theory (CT) and Higher Dimensional Algebra (HDA). Severalimportant relational structures present in organisms and the humanmind are naturally represented in terms of universal CT concepts,variable topology, non-Abelian categories and HDA-based notions.Such relatively new concepts are defined in the appropriatesequence beginning with the concept of groupoid which is fundamentalto all algebraic topology studies [63], [69], and that also turnsout to be essential to numerous applications in mathematical biology[11]-[23],[34],[74], including those of higher dimensionalgroupoids in theoretical neuroscience [38],[69]-[70].

An unifying theme of local-to-global approaches to organismal development,biological evolution and human consciousness leads to novel patternsof relations that emerge in super- and ultra- complex systems interms of global compositions of local procedures [33],[39]. Thisnovel algebraic topology concept of combination of localprocedures is suggested to be relevant to both ontogeneticdevelopment and organismal evolution, beginning with the origin ofspecies of higher organisms. Fundamentally inter–related, higherhomotopy and holonomy groupoid concepts may provide a formalframework for an improved understanding of evolutionary biology andthe origin of species on multiple levels–from molecular to speciesand biosphere levels. All key concepts pertaining to this contextare here defined for a self-contained presentation, notwithstandingthe difficulties associated with understanding the essence of life,the human mind, consciousness and its origins. One can definepragmatically the human brain in terms of its neurophysiologicalfunctions, anatomical and microscopic structure, but one cannot asreadily observe and define the much more elusive human mind whichdepends both upon a fully functional human brain and its training oreducation by the human society. Human minds that do not but weaklyinteract with those of any other member of society are partiallydisfunctional, and this creates increasing problems with the societyintegration of large groups of people that only interact weakly withall the other members of society. Obviously, it does take a fullyfunctional mind to observe and understand the human mind. It isthen claimed that human consciousness is an unique phenomenonwhich should be regarded as a composition, or combination ofultra-complex, global processes of subprocesses, at ameta-level not sub–summed by, but compatible with, humanbrain dynamics [11]–[23],[33]. Thus, a defining characteristic ofsuch conscious processes involves a combination of globalprocedures or meta-processes– such as the parallel processing ofboth image and sound sensations, perceptions and emotions, decisionmaking and learned reflexes, etc.– that ultimately leads to theontological meta-level of the ultra-complex, human mind. In thismonograph we shall not attempt to debate if other species arecapable of consciousness, or to what extent, but focus instead onthe ultra-complex problems raised by human consciousness and itsemergence. Current thinking [87], [91],[182],[186],[188], [190], [195]-[196],[203],[247] considers theactual emergence of human consciousness [83],[91],[186],[190],[261]–and also its ontic category– to be critically dependent upon theexistence of both a human society level of minimal (tribal)organization [91],[186],[190], and that of an extremely complex structural–functional unit –the human brain with an asymmetricnetwork topology and a dynamic network connectivity of veryhigh-order [187],[218], [262]. Then, an extension of the concept of coevolutionof human consciousness and society leads one to the concept of socialconsciousness [190]. One arrives also at the conclusion that the humanmind and consciousness are the result not only of theco-evolution of man and his society [91],[186],[190], but that they are, in fact, the result ofthe original co-emergence of the meta-level of a minimally-organized human society with that of several,ultra-complex human brains. Unlike the myth of only one Adam and oneEve being the required generator of human society, our co-emergenceconcept leads necessarily to the requirement of several such‘primitive’ human couples co-existing in order to generate both aminimally organized society and several, minimally self-conscious,interacting H. sapiens minds that shaped the first Rosettagroupoids of H. sapiens into human tribes. The human‘spirit’ and society are, thus, completely inseparable–justlike the very rare Siamese twins. Therefore, the appearance of humanconsciousness is considered to be critically dependent upon thesocietal co-evolution, the emergence of an elaboratelanguage-symbolic communication system, as well as the existence of‘virtual’, higher dimensional, non–commutative processes thatinvolve separate space and time perceptions in the human mind. Twofundamental, logic adjointness theorems are considered that providea logical basis for categorical representations of functional genomeand organismal networks in variable categories and extended toposes,or topoi, ‘classified’ (or encoded) by multi-valued logic algebras;their subtly nuanced connections to the variable topology andmultiple geometric structures of developing organisms are alsopointed out. Theories of the mind are thus considered in the contextof a novel ontological theory of levels. Our ultra-complexityviewpoint throws new light on previous semantic models in cognitivescience and on the theory of levels formulated within the frameworkof Categorical Ontology [40],[69]. Our novel approach tometa-systems and levels using Category Theory and HDA mathematicalrepresentations is also applicable–albeit in a modified form–tosupercomputers, complex quantum computers, man–made neural networksand novel designs of advanced artificial intelligence (AI) systems(AAIS). Anticipatory systems and complex causality at the toplevels of reality are also discussed in the context of ComplexSystems Biology (CSB), psychology, sociology and ecology. Aparadigm shift towards non-commutative, or more generally,non-Abelian theories of highly complex dynamics [33],[40],[69] issuggested to unfold now in physics, mathematics, life and cognitivesciences, thus leading to the realizations of higher dimensionalalgebras in neurosciences and psychology, as well as in humangenomics, bioinformatics and interactomics. The presence of strangeattractors in modern society dynamics, and especially the emergenceof new meta-levels of still-higher complexity in modern society,gives rise to very serious concerns for the future of mankind andthe continued persistence of a multi-stable Biosphere if suchultra-complexity, meta-level issues continue to be ignored.

Keywords: Categorical Ontology ofSuper-Complex and Ultra-Complex System Dynamics,Higher DimensionalAlgebra of Networks,Theoretical Biology and Variable Groupoids,Non-Abelian Quantum Algebraic Topology and Quantum Double Groupoids,Higher Homotopy-General van Kampen theorems; autistic children,advanced artificial intelligence and biomimetics

2000 Mathematics Subject Classification: 16B50, 68Q15.

1. Introduction

Ontology has acquired over time several meanings, and it has alsobeen approached in many different ways, but all of these areconnected to the concepts of an ‘objective existence’ andcategories of items. A related, important function of Ontology isto classify and/or categorize items and essential aspects ofreality [2],[206]-[210]. We shall employ therefore the adjective“ontological” with the meaning of pertaining to objective,real existence in its essential aspects. We shall also considerhere the noun existence as a basic, or primary concept whichcannot be defined in either simpler or atomic terms, with the latterin the sense of Wittgenstein. Furthermore, generatingmeaningful classifications of items that belong to theobjective reality is also a related, major task of ontology.Mathematicians specialised in Group Theory are also familiar withthe classification problem into various types of the mathematicalobjects called groups. Computer scientists that carry outontological classifications, or study AI and Cognitive Science [201], arealso interested in the logical foundations of computer science [1],[194],[197],[201].

For us the most interesting question by far is how humanconsciousness and civilisation emerged subsequent only to theemergence of H. sapiens. This may have arisen through thedevelopment of speech-syntactic language and an appropriatelyorganized ‘primitive’ society [91],[186] (perhaps initially made ofhominins/hominides). No doubt, the details of this highly complex,emergence process have been the subject of intense controversiesover the last several centuries, and many differing opinions, evenamong these authors, and they will continue to elude us since muchof the essential data must remains either scarce or unattainable. Itis however known that the use of cooked food, and so of fire, was necessary forthe particular physiognomy of even H. erectus, as against otherprimates, and such use perhaps required a societal context several millenia even beforethis hominin, partly in terms of the construction of hearths, which were a necessityfor the efficient cooking of food.

Other factors such as the better use of purposefully designedtools, simple weapons and the intense struggle for the survival ofthe fittest have also contributed greatly to the selectiveadvantages of H. sapiens in the fierce struggle for itsexistence; nevertheless, there is an overwhelming consensus in thespecialised literature that the co-evolution of the humanmind and society was the predominant, or key factor for the survivalof H. sapiens over that of all other closely related speciesin the genus Homo that did not survive– in spite of havingexisted earlier, and some probably much longer than H.sapiens.

The authors aim at a concise presentation of novel methodologies forstudying such difficult, as well as controversial, ontologicalproblems of Space and Time at different levels of objective realitydefined here as Complex, Super–Complex and Ultra–Complex DynamicSystems, simply in order ‘to divide and conquer’. The latter two arebiological organisms, human (and perhaps also hominide) societies,and more generally, variable ‘systems’ and meta-systems that are notrecursively–computable. Rigorous definitions of the logical andmathematical concepts employed here, as well as a step-by-stepconstruction of our conceptual framework, were provided in a recentseries of publications on categorical ontology of levels and complexsystems dynamics [33]-[34],[39]-[40]. The continuation of the veryexistence of human society may now depend upon an improvedunderstanding of highly complex systems and the human mind, and alsoupon how the global human society interacts with the rest of thebiosphere and its natural environment. It is most likely that suchtools that we shall suggest here might have value not only to thesciences of complexity and Ontology but, more generally also, to allphilosophers seriously interested in keeping on the rigorous side ofthe fence in their arguments. Following Kant’s critique of ‘pure’reason and Wittgenstein’ s critique of language misuse inphilosophy, one needs also to critically examine the possibility ofusing general and universal, mathematical language and tools informal approaches to a rigorous, formal Ontology. Throughout thismonograph we shall use the attribute ‘categorial’ only forphilosophical and linguistic arguments. On the other hand, we shallutilize the rigorous term ‘categorical’ only in conjunctionwith applications of concepts and results from the more restrictive,but still quite general, mathematical Theory of Categories,Functors and Natural Transformations (TC-FNT). According to SEP(2006): “Category theory … is a general mathematicaltheory of structures and of systems of structures.Category theory is both an interesting object of philosophicalstudy, and a potentially powerful formal tool for philosophicalinvestigations of concepts such as space, system, and even truth…It has come to occupy a central position in contemporary mathematicsand theoretical computer science, and is also applied tomathematical physics.” [248]. Traditional, modern philosophy–considered as a search for improving knowledge and wisdom– doesalso aims at unity that might be obtained as suggested by HerbertSpencer in 1862 through a ‘synthesis of syntheses’; thiscould be perhaps iterated many times because each treatment is basedupon a critical evaluation and provisional improvements of previoustreatments or stages. One notes however that this methodologicalquestion is hotly debated by modern philosophers beginning, forexample, by Descartes before Kant and Spencer; Descartes championedwith a great deal of success the ‘analytical’ approach inwhich all available evidence is, in principle, examinedcritically and skeptically first both by the proposer of novelmetaphysical claims and his, or her, readers. Descartes equated the‘synthetic’ approach with the Euclidean ‘geometric’ (axiomatic)approach, and thus relegated synthesis to a secondary, perhaps lesssignificant, role than that of critical analysis ofscientific ‘data’ input, such as the laws, principles, axioms andtheories of all specific sciences. Spinoza’s, Kant’s and Spencer’sstyles might be considered to be synthetic by Descartes and allCartesians, whereas Russell’s approach might also be considered tobe analytical. Clearly and correctly, however, Descartes did notregard analysis ( $A$ ) and synthesis ( $S$ ) as exactly inverse to eachother, such as $A\\rightleftarrows S$ , and also not merely as‘bottom–up’ and ‘top–bottom’ processes ( $\\downarrow\\uparrow$ ).Interestingly, unlike Descartes’ discourse of the philosophicalmethod, his treatise of philosophical principles comes closer to thesynthetic approach in having definitions and deductive attempts,logical inferences, not unlike his ‘synthetic’ predecessors, albeitwith completely different claims and perhaps a wider horizon. Thereader may immediately note that if one, as proposed by Descartes,begins the presentation or method with an analysis $A$ , followed bya synthesis S, and then reversed the presentation in a follow-uptreatment by beginning with a synthesis $S*$ followed by an analysis $A^{\\prime}$ of the predictions made by $S^{\\prime}$ consistent, or analogous, with $A$ , then obviously $AS\eq S^{\\prime}A^{\\prime}$ because we assumed that $A\\simeq A^{\\prime}$ and that $S\eq S^{\\prime}$ . Furthermore, if one did not make anyadditional assumptions about analysis and synthesis, then $analysis\\rightarrow synthesis\eq synthesis\\rightarrow analysis$ , or $AS\eq SA$ , that is analysis and synthesis obviously ‘do notcommute’; such a theory when expressed mathematically would be thencalled ‘non-Abelian’. This is also a good example of themeaning of the term non-Abelian in a philosophical, epistemologicalcontext.

2.The Theory of Levels in Categorial and Categorical Ontology

This section outlines our novel methodology and approach to theontological theory of levels, which is then applied in subsequentsections in a manner consistent with our recently publisheddevelopments [33]-[34],[39]-[40]. Here, we are in harmony with thetheme and approach of Poli’s ontological theory of levels of reality[121], [206]–[211]) by considering both philosophical–categorialaspects such as Kant’s relational and modal categories, as well ascategorical–mathematical tools and models of complex systems interms of a dynamic, evolutionary viewpoint.

We are then presenting a Categorical Ontology of highly complexsystems, discussing the modalities and possible operational logicsof living organisms, in general. Then, we consider briefly thoseintegrated functions of the human brain that support theultra-complex human mind and its important roles in societies. Moresspecifically, we propose to combine a critical analysis of languagewith precisely defined, abstract categorical concepts from AlgebraicTopology reported by Brown et al, in 2007 [69], and thegeneral-mathematical Theory of Categories, Functors and NaturalTransformations: [56], [80], [98]-[102],[105]-[106],[113],[115-[119],[130], [133]-[135],[141]-[143],[151],[154], [161]-[163],[165]-[168], [172], [175]-[177],[183],[192]-[194],[198]-[199] [213]-[215],[225], [227],[246], [252], [256]into a categorical framework which is suitable for furtherontological development, especially in the relational rather thanmodal ontology of complex spacetime structures. Basic concepts ofCategorical Ontology are presented in this section, whereas formaldefinitions are relegated to one of our recent, detailed reports[69]. On the one hand, philosophical categories according to Kantare: quantity, quality, relation and modality, and themost complex and far-reaching questions concern the relational andmodality-related categories. On the other hand, mathematicalcategories are considered at present as the most general anduniversal structures in mathematics, consisting of relatedabstract objects connected by arrows. The abstract objects ina category may, or may not, have a specified structure, butmust all be of the same type or kind in any given category. Thearrows (also called ‘morphisms’) can represent relations,mappings/functions, operators, transformations, homeomorphisms, andso on, thus allowing great flexibility in applications, includingthose outside mathematics as in: Logics [118]-[120], ComputerScience [1], [161]-[163] [201],[248], [252], Life Sciences[5],[11]-[17],[19],[23],[28]-[36],[39],[40],[42]-[44],[70],[74],[103]-[104],[230],[232],[234]-[238],[264],Psychology, Sociology [33],[34],[39],[40],[74], and EnvironmentalSciences [169]. The mathematical category also has a form of‘internal’ symmetry, specified precisely as thecommutativity of chains of morphism compositions that areuni-directional only, or as naturality of diagrams ofmorphisms; finally, any object A of an abstract category has anassociated, unique, identity, $1_{A}$ , and therefore, one can replaceall objects in abstract categories by the identity morphisms. Whenall arrows are invertible, the special category thus obtainedis called a ‘groupoid’, and plays a fundamental role in thefield of mathematics called Algebraic Topology.

The categorical viewpoint– as emphasized by William Lawvere,Charles Ehresmann and most mathematicians– is that the key conceptand mathematical structure is that of morphisms that can beseen, for example, as abstract relations, mappings, functions,connections, interactions, transformations, and so on. Thus, onenotes here how the philosophical category of ‘relation’ isclosely allied to the basic concept of morphism, or arrow, in anabstract category; the implicit tenet is that arrows are whatcounts. One can therefore express all essential properties,attributes, and structures by means of arrows that, in the mostgeneral case, can represent either philosophical ‘relations’ ormodalities, the question then remaining if philosophical–categorialproperties need be subjected to the categorical restriction ofcommutativity. As there is no a priori reason ineither nature or ‘pure’ reasoning, including any form of Kantian‘transcendental logic’, that either relational or modal categoriesshould in general have any symmetry properties, one cannot imposeonto philosophy, and especially in ontology, all the strictures ofcategory theory, and especially commutativity. Interestingly, thesame comment applies to Logics: only the simplest forms of Logics,the Boolean and intuitionistic, Heyting-Brouwer logic algebras arecommutative, whereas the algebras of many-valued (MV) logics, suchas Łukasiewicz logic are non-commutative (ornon-Abelian).

3. Basic Structure of Categorical Ontology.

The Theory of Levels: Emergence of Higher Levels, Meta–Levels and Their Sublevels

With the provisos specified above, our proposed methodology andapproach employs concepts and mathematical techniques from CategoryTheory which afford describing the characteristics and binding ofontological levels besides their links with other theories. WhereasHartmann in 1952 stratified levels in terms of the four frameworks:physical, ‘organic’/biological, mental and spiritual [137], werestrict here mainly to the first three. The categorical techniqueswhich we introduce provide a powerful means for describing levels inboth a linear and interwoven fashion, thus leading to the necessarybill of fare: emergence, complexity and open non-equilibrium, orirreversible systems. Furthermore, any effective approach toPhilosophical Ontology is concerned with universal itemsassembled in categories of objects and relations, involving, ingeneral, transformations and/or processes. Thus, CategoricalOntology is fundamentally dependent upon both space and timeconsiderations. Therefore, one needs to consider first a dynamicclassification of systems into different levels of reality,beginning with the physical levels (including the fundamentalquantum level) and continuing in an increasing order of complexityto the chemical–molecular levels, and then higher, towards thebiological, psychological, societal and environmental levels.Indeed, it is the principal tenet in the theory of levels that :“there is a two-way interaction between social and mentalsystems that impinges upon the material realm for which the latteris the bearer of both” [209]. Therefore, any effective CategoricalOntology approach requires, or generates–in the constructivesense–a ‘structure’ or pattern of linkeditems rather than a discrete set of items. The evolution in ouruniverse is thus seen to proceed from the level of ‘elementary’quantum ‘wave–particles’, their interactions via quantizedfields (photons, bosons, gluons, etc.), also including the quantumgravitation level, towards aggregates or categories of increasingcomplexity. In this sense, the classical macroscopic systems aredefined as ‘simple’ dynamical systems that are computablerecursively as numerical solutions of mathematical systems ofeither ordinary or partial differential equations. Underlying suchmathematical systems is always the Boolean, or chrysippian, logic,namely, the logic of sets, Venn diagrams, digital computers andperhaps automatic reflex movements/motor actions of animals. Thesimple dynamical systems are always recursively computable (see forexample, Suppes, 1995–2006 [253]-[254], and also [23]), and in acertain specific sense, both degenerate and non-generic, andconsequently also they are structurally unstable to smallperturbations; such systems are, in general, deterministic in theclassical sense, although there are arguments about the possibilityof chaos in quantum systems. The next higher order of systems isthen exemplified by ‘systems with chaotic dynamics’ that areconventionally called ‘complex’ by physicists who study ‘chaotic’dynamics/Chaos theories, computer scientists and modelers eventhough such physical, dynamical systems are still completelydeterministic. It has been formally proven that such ‘systems withchaos’ are recursively non-computable (see for example, refs.[23] and [28] for a 2-page, rigorous mathematical proof andrelevant references), and therefore they cannot be completely andcorrectly simulated by digital computers, even though some are oftenexpressed mathematically in terms of iterated maps oralgorithmic-style formulas. Higher level systems above the chaoticones, that we shall call ‘Super–Complex, Biologicalsystems’, are the living organisms, followed at still higher levelsby the ultra-complex ‘systems’ of the human mind and humansocieties that will be discussed in the last sections. The evolutionto the highest order of complexity- the ultra-complex,meta–‘system’ of processes of the human mind–may have becomepossible, and indeed accelerated, only through human societalinteractions and effective, elaborate/rational and symboliccommunication through speech (rather than screech –as in the caseof chimpanzees, gorillas, baboons, etc).

4. Fundamental Concepts of Algebraic Topology with Potential Application toOntology Levels Theory and the Classification of SpaceTime Structures

We shall consider in this section the potential impact of novelAlgebraic Topology concepts, methods and results on the problems ofdefining and classifying rigorously Quantum Spacetimes (QSS)[3],[36]-[38],[69], [78]-[79]. The 600-page project manuscript,‘Pursuing Stacks’ written by Alexander Grothendieck in 1983was partly aimed at a non-Abelian homological algebra; it didnot achieve this goal but has been very influential in thedevelopment of weak $n$ -categories and other highercategorical structures that are relevant to QSS structures. Withthe advent of Quantum Groupoids–generalizing Quantum Groups,Quantum Algebra and Quantum Algebraic Topology, several fundamentalconcepts and new theorems of Algebraic Topology may also acquire anincreased importance through their potential applications to currentproblems in theoretical and mathematical physics, such as thosedescribed in an available preprint [38], and also in several otherrecent publications [36]–[37], [69].In such novel applications, both the internal and external groupoidsymmetries [265] may too acquire new physical significance. Thus, ifquantum theories were to reject the notion of a continuummodel for spacetime, then it would also have to reject the notion ofthe real line and the notion of a path. How then is one to constructa homotopy theory? One possibility is to take the route signalled byČech [82], and which later developed in the hands of Borsuk into‘Shape Theory’ [86]. Thus a quite general space is studiedindirectly by means of its approximation by open covers. Yet anotherpossible approach is briefly outlined in the next section.

Several fundamental concepts of Algebraic Topology and CategoryTheory that are needed throughout this monograph will be introducednext so that we can reach an extremely wide range of applicability,especially to the higher complexity levels of reality. Fullmathematical details are also available in a recent paper by Brownet al. [69] that focused on a mathematical–conceptual frameworkfor a formal approach to Categorical Ontology and the Theory ofOntological Levels [206], [40].

Groupoids, Topological Groupoids, Groupoid Atlases and Locally Lie Groupoids

Recall that a groupoid $\\mathsf{G}$ is a small category in which every morphism is an isomorphism.

Topological Groupoids

An especially interesting concept is that of a topologicalgroupoid which is a groupoid internal to the category $\\mathsf{Top}$ ; further mathematical details are presented in thepaper by Brown et al. in 2007 [69].

Groupoid Atlases

Motivation for the notion of a groupoid atlas comes from consideringfamilies of group actions, in the first instance on the same set. Asa notable instance, a subgroup $H$ of a group $G$ gives rise to agroup action of $H$ on $G$ whose orbits are the cosets of $H$ in $G$ . However a common situation is to have more than one subgroup of $G$ , and then the various actions of these subgroups on $G$ arerelated to the actions of the intersections of the subgroups. Thissituation is handled by the notion of global action, asdefined in [41]. A key point in this construction is that the orbitsof a group action then become the connected components of agroupoid. Also this enables relations with other uses of groupoids.The above account motivates the following. A groupoid atlas $\\mathcal{A}$ on a set $X_{\\mathcal{A}}$ consists of a family of ‘local groupoids’ $(\\mathsf{G}_{\\mathcal{A}})$ defined with respective object sets $(X_{\\mathcal{A}})_{\\alpha}$ taken to be subsets of $X_{\\mathcal{A}}$ . These local groupoids are indexed bya set $\\Psi_{\\mathcal{A}}$ , again called the coordinate system of $\\mathcal{A}$ which is equipped with a reflexive relation denoted by ${\\leqslant}$ . Thisdata is to satisfy several conditions reported in [41] by Bak etal. in 2006, and also discussed in [63] in the context ofCategorical Ontology.

The van Kampen Theorem and Its Generalisations to Groupoids and Higher Homotopy

The van Kampen Theorem has an important and also anomalous rôlein algebraic topology. It allows computation of an importantinvariant for spaces built up out of simpler ones. It is anomalousbecause it deals with a non-Abelian invariant, and has not been seenas having higher dimensional analogues. However, Brown found in 1967a generalisation of this theorem to groupoids [60], stated asfollows. In this, $\\pi_{1}(X,X_{0})$ is the fundamental groupoidof $X$ on a set $X_{0}$ of base points: so it consists of homotopyclasses rel end points of paths in $X$ joining points of $X_{0}\\cap X$ . Such methods were extended successfully by R. Brown tohigher dimensions. The potential applications of the Higher Homotopyvan Kampen Theorem [37]-38] were already discussed in a previous paper [69] publishedby Brown, Glazebrook and Baianu in 2007.

5. Local-to-Global Problems in Spacetime Structures. Symmetry Breaking,Irreversibility and the Emergence of Highly Complex Dynamics

Spacetime Local Inhomogeneity, Discreteness and Broken Symmetries: From Local to Global Structures.

On summarizing in this section the evolution of the physicalconcepts of space and time, we are pointing out first how the viewschanged from homogeneity and continuity to inhomogeneity anddiscreteness. Then, we link this paradigm shift to a possible,novel solution in terms of local-to-global approaches and proceduresto spacetime structures. These local-to-global procedures procedureswill therefore lead to a wide range of applications sketched in thelater sections, such as the emergence of higher dimensionalspacetime structures through highly complex dynamics in organismicdevelopment, adaptation, evolution, consciousness and societyinteractions.

Classical physics, including GR involves a concept of bothcontinuous and homogeneous space and time with strictcausal (mechanistic) evolution of all physical processes(“God does not play dice”, cf. Albert Einstein).Furthermore, up to the introduction of quanta–discreteportions, or packets–of energy by Ernst Planck (which was furtherelaborated by Einstein, Heisenberg, Dirac, Feynman, Weyl and othereminent physicists of the last century), energy was also consideredto be a continuous function, though not homogeneously distributed inspace and time. Einstein’s Relativity theories joined together spaceand time into one ‘new’ entity–the concept of spacetime. Inthe improved form of GR, inhomogeneities caused by the presence ofmatter are also allowed to occur in spacetime. Causality, however,remained strict, but also more complicated than in theNewtonian theories as discontinuities appear in spacetime in theform of singularities, or ‘black holes. The standard GR theory, theMaxwellian Theory of Electromagnetism and Newtonian mechanics canall be considered Abelian, even though GR not only allows,but indeed, requires spacetime inhomogeneities to occur in thepresence of gravitational fields, unlike Newtonian mechanicswhere space is both absolute and homogeneous. Recent effortsto develop non-Abelian GR theories–especially with an intent todevelop Quantum Gravity theories– seem to have considered bothpossibilities of locally homogeneous or inhomogeneous, but stillglobally continuous spacetimes. The successes of non-Abelian gaugetheories have become well known in physics since 1999, but they still await theexperimental discovery of their predicted Higgsboson particles [267].

Although Einstein’s Relativity theories incorporate the concept ofquantum of energy, or photon, into their basic structures,they also deny such discreteness to spacetime even though thediscreteness of energy is obviously accepted within Relativitytheories. The GR concept of spacetime being modified, ordistorted/‘bent’, by matter goes further back to Riemann, butit was Einstein’s GR theory that introduced the idea of representinggravitation as the result of spacetime distortion by matter.Implicitly, such spacetime distortions remained continuous eventhough the gravitational field energy –as all energy– was allowedto vary in discrete, albeit very tiny portions–thegravitational quanta. So far, however, the detection of gravitons–the quanta of gravity–related to the spacetime distortions bymatter–has been unsuccessful. Mathematically elegant/precise andphysically ‘validated’ through several crucial experiments andastrophysical observations, Einstein’s GR is obviously notreconcilable with Quantum theories (QTs). GR was designed as thelarge–scale theory of the Universe, whereas Quantumtheories–at least in the beginning–were designed to address theproblems of microphysical measurements at very tiny scales ofspace and time involving extremely small quanta of energy. We seetherefore the QTs vs. GR as a local-to-global problem that has notbeen yet resolved in the form of an universally valid QuantumGravity. Promising, partial solutions are suggested in three recentpapers [36],[38], [70]. Quantum theories (QTs) were developed thatare just as elegant mathematically as GR, and they were alsophysically ‘validated’ through numerous, extremely sensitive andcarefully designed experiments.

However, to date quantum theories have not yet been extended, orgeneralized, to a form capable of recovering the results ofEinstein’s GR as a quantum field theory over a GR-spacetime alteredby gravity. Furthermore, quantum symmetries occur not only onmicrophysical scales, but also macroscopically in certain, ‘special’cases, such as liquid ${}^{3}$ He close to absolute zero andsuperconductors where extended coherence is possible for thesuperfluid, long-range coupled Cooper electron-pairs. However,explaining such interesting physical phenomena also requires theconsideration of symmetry breaking resulting from theGoldstone Boson Theorem as it was shown in [267].Occasionally, symmetry breaking is also invoked in the recentscience literature as a ‘possible mechanism for human consciousness’which also seems to be related to, or associated with some form of‘global coherence’–over most of the brain; however, the existenceof such a ‘quantum coherence in the brain’–at least atphysiological temperatures–as it would be preciselyrequired/defined by QTs, is a most unlikely event. On the otherhand, a quantum symmetry breaking in a neural networkconsidered metaphorically as a Hopfield (‘spin-glass’) network mightbe conceivable close to physiological temperatures, except for thelack of evidence of the existence of any requisite (electron) spinlattice structure that is indeed an absolute requirement in such aspin-glass metaphor.

Now comes the real, and very interesting part of the story: neuronalnetworks do form functional patterns and structures that possesspartially ‘broken’, or more general symmetries than those describedby quantum groups. Such extended symmetries can bemathematically determined, or specified, by certaingroupoids–that were previously called‘neuro-groupoids’ [33]. Even more generally, genetic networks alsoexhibit extended symmetries that are present in biological species which are represented bya biogroupoid structure, as previously defined and discussedby Baianu, Brown, Georgescu and Glazebrook in [32]-[33]. Suchbiogroupoid structures [33] can be experimentally validated, for example,at least partially through Functional Genomics observations andcomputer, bioinformatics processing [30]. We shall discuss furtherseveral such interesting groupoid structures in the followingsections, and also how they have already been utilized in so-called‘local-to-global procedures’ in order to construct ‘global’solutions; such global solutions in quite complex (holonomy) casescan still be unique up to an isomorphism (theGlobalisation Theorem, as it was discussed in [69], and referencescited therein. Last-but-not-least, holonomy may provide aglobal solution, or ‘explanation for memory storage by‘neuro-groupoids’. Uniqueness holonomy theorems might possiblyexplain the existence of unique, persistent and resilient memories.

Towards Biological Postulates and Principles

Whereas the hierarchical theory of levels provides a powerful,systems approach through categorical ontology, the foundation ofscience involves universal models and theories pertaining todifferent levels of reality. It would seem natural to expect thattheories aimed at different ontological levels of reality shouldhave different principles. We are advocating the need for developingprecise, but nevertheless ‘flexible’, concepts and novelmathematical representations suitable for understanding theemergence of the higher complexity levels of reality. Such theoriesare based on axioms, principles, postulates and laws operating ondistinct levels of reality with a specific degree of complexity.Because of such distinctions, inter-level principles or laws arerare and over-simplified principles abound. Alternative approachesmay be, however, possible based upon an improved ontological theoryof levels. Interestingly, the founder of Relational Biology, NicolasRashevsky proposed in 1969 that physical laws and principles can beexpressed in terms of mathematical functions, or mappings,and are thus being predominantly expressed in a numericalform, whereas the laws and principles of biological organisms andsocieties need take a more general form in terms of quite general,or abstract–mathematical and logical relations which cannot alwaysbe expressed numerically; the latter are often qualitative, whereasthe former are predominantly quantitative [224].

Rashevsky focused his Relational Biology/Society Organization paperson a search for more general relations in Biology and Sociology thatare also compatible with the former. Furthermore, Rashevsky proposedtwo biological principles that add to Darwin’s natural selection andthe ‘survival of the fittest principle’, the emergentrelational structure that are defining the adaptive organism:

1. The Principle of Optimal Design[233],
and

2. The Principle of Relational Invariance (initially phrased by
Rashevsky as “Biological Epimorphism”)[12]-[13],[15],[222].

In essence, the ‘Principle of Optimal Design’ [233] defines theorganization and structure of the ‘fittest’ organism which survivesin the natural selection process of competition between species, interms of an extremal criterion, similar to that of Maupertuis; theoptimally ‘designed’ organism is that which acquires maximumfunctionality essential to survival of the successful species at thelowest ‘cost’ possible [11]-[13]. The ‘design’ in this case is commonly takenin the sense of the result of a long evolutionary process thatoccurred under various environmental and propagation constraints orselection ‘pressures’, such as that caused by sexual reproduction inDarwin’s model of the origin of species during biological evolution.The ‘costs’ are here defined in the context of the environmentalniche in terms of material, energy, genetic and organismic processesrequired to produce/entail the pre-requisite biological function(s)and their supporting anatomical structure(s) needed for competitivesurvival in the selected niche. Further details were presented byRobert Rosen in his short, but significant, book on optimalityprinciples in theoretical biology [233], published in 1967.

The ‘Principle of Biological Epimorphism’, on the other hand, statesthat the highly specialized biological functions of higher organismscan be mapped (through an epimorphism) onto those of the simplerorganisms, and ultimately onto those of a (hypothetical) primordialorganism (which is assumed to be unique up to an isomorphism orselection-equivalence). The latter proposition, as formulatedby Rashevsky, is more akin to a postulate than a principle. However,it was then generalised and re-stated as the Postulate of RelationalInvariance [12]. Somewhat similarly, a dual principle and thecolimit construction were invoked for the ontogenetic development oforganisms [11], and more recently other quite similar colimitconstructions were considered in relation to ‘Memory EvolutiveSystems’, or phylogeny [103]-[104].

An axiomatic system (ETAS) leading to higher dimensional algebras oforganisms in supercategories has also been formulated [18] whichspecifies both the logical and the mathematical ( $\\pi-$ ) structuresrequired for complete self-reproduction and self-reference,self-awareness, etc. of living organisms. To date, there is nohigher dimensional algebra (HDA) axiomatics other than the ETASproposed for complete self-reproduction in super-complex systems, orfor self-reference in ultra-complex ones. On the other hand, thepreceding, simpler ETAC axiomatics introduced by Lawvere, wasproposed for the foundation of ‘all’ mathematics, includingcategories [166]-[167], but this seems to have occurred before theactual emergence of HDA.

6. Towards a Formal Theory of Levels in Ontology

This subsection will introduce in a concise manner fundamentalconcepts of the ontological theory of levels. Further details werereported by Poli in [206]-[211], and by Baianu and Poli in thisvolume [40].

Fundamentals of Poli’s Theory of Levels

The ontological theory of levels by Poli [206]-[211] considers ahierarchy of items structured on different levels of reality,or existence, with the higher levels emerging from the lower,but usually not reducible to the latter, as claimed bywidespread reductionism.This approach modifies and expands considerably earlier work byHartmann [137] both in its vision and the range of possibilities.Thus, Poli in [206]-[211] considers four realms or levels ofreality: Material-inanimate/Physico-chemical,Material-living/Biological, Psychological and Social. Poli in [211]has stressed a need for understanding causal andspatiotemporal phenomena formulated within a descriptivecategorical context for theoretical levels of reality. There is theneed in this context to develop a synthetic methodology inorder to compensate for the critical ontic data analysis, althoughone notes (cf. Rosen in 1987 [232]) that analysis and synthesis arenot the exact inverse of each other. At the same time, we address incategorical form the internal dynamics, the temporalrhythm, or cycles, and the subsequent unfolding of reality. Thegenera of corresponding concepts such as ‘processes’, ‘groups’,‘essence’, ‘stereotypes’, and so on, can be simply referred to as‘items’ which allow for the existence of many forms of causalconnection [210]-[211]. The implicit meaning is that the irreduciblemultiplicity of such connections converges, or it is ontologicallyintegrated within a unified synthesis.

The Object-based Approach vs Process-based (Dynamic) Ontology

In classifications, such as those developed over time in Biology fororganisms, or in Chemistry for chemical elements, the objectsare the basic items being classified even if the ‘ultimate’ goal maybe, for example, either evolutionary or mechanistic studies. Anontology based strictly on object classification may have little tooffer from the point of view of its cognitive content. It isinteresting that D’Arcy W. Thompson arrived in 1941 at an ontologic“principle of discontinuity” which “is inherent in all ourclassifications, whether mathematical, physical or biological… Inshort, nature proceeds from one type to another among organicas well as inorganic forms… and to seek for stepping stones acrossthe gaps between is to seek in vain, for ever.” (p.1094 of Thompsonin [259], re-printed edition). Whereas the existence of differentontological levels of reality is well-established, one cannot alsodiscard the study of emergence and co-emergence processes as a pathto improving our understanding of the relationships among theontological levels, and also as an important means of ontologicalclassification. Furthermore, the emergence of ontologicalmeta-levels cannot be conceived in the absence of the simplerlevels, much the same way as the chemical properties of elements andmolecules cannot be properly understood without those of theirconstituent electrons.

It is often thought that the object-oriented approach can bereadily converted into a process-based one. It would seem, however,that the answer to this question depends critically on theontological level selected. For example, at the quantum level,object and process become inter-mingled. Either comparing ormoving between levels– for example through emergent processes–requires ultimately a process-based approach, especially inCategorical Ontology where relations and inter-process connectionsare essential to developing any valid theory. Ontologically, thequantum level is a fundamentally important starting point whichneeds to be taken into account by any theory of levels that aims atcompleteness. Such completeness may not be attainable, however,simply because an ‘extension’ of Gödel’s theorem may hold herealso. The fundamental quantum level is generally accepted to bedynamically, or intrinsically non-commutative, in the senseof the non-commutative quantum logic and also in the sense ofnon-commuting quantum operators for the essential quantumobservables such as position and momentum. Therefore, anycomprehensive theory of levels, in the sense of incorporating thequantum level, is thus –mutatis mutandis– non-Abelian. A paradigm shift towards a non-AbelianCategorical Ontology has already begun [33]-[34],[37]-[38],[40],[69].

From Component Objects and Molecular/Anatomical Structure to Organismic Functions andRelations: A Process–Based Approach to Ontology

Wiener in 1950 made the important remark that implementation ofcomplex functionality in a (complicated, but not necessarilycomplex–in the sense defined above) machine requires also thedesign and construction of a correspondingly complexstructure, or structures [269]. A similar argument holdsmutatis mutandis, or by induction, for variablemachines, variable automata and variable dynamic systems [12]-[23];therefore, if one represents organisms as variable dynamic systems,one a fortiori requires a super-complex structure toenable or entail super-complex dynamics, and indeed this isthe case for organisms with their extremely intricate structures atboth the molecular and supra-molecular levels. This seems tobe a key point which appears to have been missed in the early-stagesof Robert Rosen’s theory of simple $(M,R)$ -systems, prior to 1970,that were deliberately designed to have “no structure” as it wasthought they would thus attain the highest degree of generality orabstraction, but were then shown by Warner to be equivalent to aspecial type of sequential machine or classical automaton [17],[264].

The essential properties that define the super– and ultra– complexsystems derive from the interactions, relations and dynamictransformations that are ubiquitous at such levels of reality–which need to be distinguished from the levels of organizationinternal to any biological organism or biosystem. Therefore, acomplete approach to Ontology should obviously includerelations and interconnections between items, with theemphasis on dynamic processes, complexity andfunctionality of systems. This leads one to consider generalrelations, such as morphisms on different levels, and thus tothe categorical viewpoint of Ontology. Theprocess-based approach to an Universal Ontology is thereforeessential to an understanding of the Ontology of Reality Levels,hierarchies, complexity, anticipatory systems, Life, Consciousnessand the Universe(s). On the other hand, the opposite approach, basedon objects, is perhaps useful only at the initial cognitive stagesin experimental science, such as the simpler classification systemsused for efficiently organizing data and providing a simple datastructure. We note here also the distinct meaning of ‘object’ inpsychology, which is much different from the one considered in thissubsection; for example, an external process can be ‘reflected’ inone’s mind as an ‘object of study’. This duality, or complementaritybetween ‘object’ and ‘subject’, ‘objective’ and ‘subjective’ seemsto be widely adopted in philosophy, beginning with Descartes andcontinuing with Kant, Heidegger, and so on. A somewhat similar, butnot precisely analogous distinction is fundamental in standardQuantum Theory– the distinction between the observed/measuredsystem (which is the quantum, ‘subject’ of the measurement ), andthe measuring instrument (which is a classical ‘object’ that carriesout the measurement).

Physicochemical Structure–Function Relationships

It is generally accepted at present that structure-functionalityrelationships are key to the understanding of super-complex systemssuch as living cells and organisms. Integrating structure–functionrelationships into a Categorical Ontology approach is undoubtedly aviable alternative to any level reduction, andphilosophical/epistemologic reductionism in general. Such anapproach is also essential to the science of complex/super-complexsystems; it is also considerably more difficult than eitherphysicalist reductionism, entirely abstract relationalism or‘rhetorical mathematics’. Moreover, because there are manyalternative ways in which the physico-chemical structures can becombined within an organizational map or relational complex system,there is a multiplicity of ‘solutions’ or mathematical modelsthat needs be investigated, and the latter are not computable with adigital computer in the case of complex/super-complex systems suchas organisms [23],[232].The problem is further compounded by the presence ofstructural disorder (in the physical structure sense) whichleads to a very high multiplicity ofdynamical-physicochemical structures (or ‘configurations’) of abiopolymer– such as a protein, enzyme, or nucleic acid, of abiomembrane, as well as of a living cell, that correspond to asingle function or a small number of physiological functions [20];this complicates the assignment of a ‘fuzzy’ physico-chemicalstructure to a well-defined biological function unless extensiveexperimental data are available, as for example, those derivedthrough computation from 2D-NMR spectroscopy data (as for example byWütrich, in 1996 [271]), or neutron/X-ray scattering and relatedmulti-nuclear NMR spectroscopy/relaxation data [20]Detailed considerations of the ubiquitous, or universal, partialdisorder effects on the structure-functionality relationships werereported for the first time by Baianu in 1980 [20]. Specific aspectswere also recently discussed by Wütrich in 1996 on the basis of2D-NMR analysis of ‘small’ protein configurations in solution [271].

As befitting the situation, there are devised universalcategories of reality in its entirety, and also subcategories whichapply to the respective sub-domains of reality. We harmonize thistheme by considering categorical models of complex systems in termsof an evolutionary dynamic viewpoint using the mathematical methodsof Category Theory which afford describing the characteristics,classification and emergence of levels, besides the links with othertheories that are, a priori, essential requirements of anyontological theory. We also underscore a significant component ofthis essay that relates the ontology to geometry/topology;specifically, if a level is defined via ‘iterates of localprocedures’ (cf ‘items in iteration’ cf. Brown and $\\dot{\\rm I}$ çen in [71]), that will further expanded upon in the lastsections; then we will have a handle on describing its intrinsicgoverning dynamics (with feedback). As we shall see in the nextsubsection, categorical techniques– which form an integral part ofour ontological considerations– provide a means of describing ahierarchy of levels in both a linear and interwoven, orentangled, fashion, thus leading to the necessary bill offare: emergence, higher complexity and open,non-equilibrium/irreversible systems. We must emphasize that thecategorical methodology selected here is intrinsically ‘higherdimensional’, and can thus account for meta–levels, such as‘processes between processes…’ within, or between, the levels–andsub-levels– in question. Whereas a strictly Boolean classificationof levels allows only for the occurrence of discreteontological levels, and also does not readily accommodate eithercontingent or stochastic sub-levels, the LM-logicalgebra is readily extended to continuous, contingentor even fuzzy sub-levels, or levels of reality [11],[23],[32]-[34],[39]-[40],[120],[140].Clearly, a Non-Abelian Ontology of Levels would require theinclusion of either Q- or LM- logics algebraic categories (discussedin the following section) because it begins at the fundamentalquantum level –where Q-logic reigns– and ‘rises’ to the emergentultra-complex level(s) with ‘all’ of its possible sub-levelsrepresented by certain LM-logics. (Further considerations on themeta–level question are presented by Baianu and Poli in this volume[40]). On each level of the ontological hierarchy there is asignificant amount of connectivity through inter-dependence,interactions or general relations often giving rise to complexpatterns that are not readily analyzed by partitioning or throughstochastic methods as they are neither simple, nor are they randomconnections. This ontological situation gives rise to a wide varietyof networks, graphs, and/or mathematical categories, all withdifferent connectivity rules, different types of activities, andalso a hierarchy of super-networks of networks of subnetworks. Then,the important question arises what types of basic symmetry orpatterns such super-networks of items can have, and how do theeffects of their sub-networks percolate through the various levels.From the categorical viewpoint, these are of two basic types: theyare either commutative or non-commutative, where, atleast at the quantum level, the latter takes precedence over theformer, as we shall further discuss and explain in the followingsections.