A conceptual construction of complexity levels theory in spacetime categorical ontology: Non-Abelian algebraic topology, many-valued logics and dynamic systems [Book Review]

Axiomathes 17 (3-4):409-493 (2007)
Abstract
A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics.
Keywords Categorical ontology and the theory of levels  Formal foundation and relational structure of categorical ontology and emergent complexity theories  Ontological essence and Universal properties of items  Mathematical categories, Groupoids, Locally Lie groupoids, Groupoid Atlas, Stacks, Fibred categories  Relational biology principles  Higher homotopy—General Van Kampen Theorems (HHvKT) and Non-Abelian algebraic topology (NAAT)  Non-commutativity of diagrams and non-Abelian theories—Non-Abelian categorical ontology  Non-commutative topological invariants of complex dynamic state spaces  Natural transformations in molecular and relational biology: Molecular class variables (mcv)  Natural transformations and the Yoneda-Grothendieck lemma/construction  Variable groupoids, Variable categories, Variable topology and atlas structures  Biomolecular classes and Metabolic repair systems
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