PlanetPhysics/Categories and Supercategories in Relational Biology

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This topic entry introduces one of the most general mathematical models of living organisms called `[[../SuperCategory6/|organismic supercategories]]' ([[../OS/|OS]]) which can be axiomatically defined to include both complete self-reproduction of logically defined π-entities founded in Quine's logic and [[../GenericityInOpenSystems/|dynamic system]] [[../TrivialGroupoid/|diagrams]] subject to both [[../CoIntersections/|algebraic]] and [[../CoIntersections/|topological]] transformations.

Organismic Supercategories (OS)

OS mathematical models were introduced as structures in [[../HigherDimensionalAlgebra2/|higher dimensional algebra]] that are mathematical interpretations of the axioms in ETAS- a natural extension of Lawvere's elementarytheoryofabstractcategories ([[../ETACAxioms/|ETAC]]) to [[../AbelianCategory3/|non-Abelian]] structures and heterofunctors.

When regarded as categorical models of supercomplex [[../NewtonianMechanics/|dynamics]] in living organisms OS provide a unified conceptual framework for [[../RSystemsCategory/|relational biology]] that utilizes flexible, algebraic and [[../TrivialGroupoid/|topological structures]] which transform naturally under heteromorphisms or heterofunctors. One of the advantages of the [[../ETACAxioms/|ETAS]] axiomatic approach, which was inspired by the [[../Work/|work]] of Lawvere (1963, 1966), is that ETAS avoids all the antimonies/paradoxes previously reported for sets (Russell and Whitehead, 1925, and Russell, 1937). ETAS also provides an axiomatic approach to recent higher dimensional algebra applications to [[../SystemsBiology/|complex systems biology]] ([1], [2] and references cited therein.)

Selected Examples of OS Applications to Relational and Complex Systems Biology

Whereas super-categories are usually defined as [[../InfinityGroupoid/|n-categories]] or in higher dimensional algebra, organismic supercategories have flexible, algebraic and topological structures that transform naturally under heteromorphisms or heterofunctors. Different approaches to relational biology and biodynamics, developed by [[../NicolasRashevsky/|Nicolas Rashevsky]], [[../RobertRosen/|Robert Rosen]] and by the author, are compared with the classical approach to qualitative dynamics of [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] (QDS). [[../VariableCategory2/|Natural transformations]] of heterofunctors in organismic supercategories lead to specific modular models of a variety of specific life processes involving dynamics of genetic systems, ontogenetic development, fertilization, regeneration, neoplasia and oncogenesis. Axiomatic definitions of [[../Cod/|categories]] and [[../OS/|supercategories]] of complex biological systems allow for dynamic [[../LQG2/|computations]] of cell transformations, neoplasia and cancer.

All Sources

[3] [4] [5] [6] [7] [8] [9] [10] [11] [1] [2] [12] [13] [14]

References

  1. ↑ 1.0 1.1 Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R) --Systems and Their Higher Dimensional Algebra; PDF of Abstract and Preprint of Report
  2. ↑ 2.0 2.1 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, Axiomathes 16 Nos. 1--2, 65--122.
  3. ↑ Bacon, John, 1985, "The completeness of a predicate--functor logic," Journal of Symbolic Logic 50: 903--926.
  4. ↑ Paul Bernays, 1959, "Uber eine naturliche Erweiterung des Relationenkalkuls." in Heyting, A., ed., Constructivity in Mathematics. North Holland: 1--14.
  5. ↑ References [14] to [34] in the "bibliography of category theory and algebraic topology"
  6. ↑ I. C. Baianu, J. F. Glazebrook, R. Brown and G. Georgescu.: Complex Nonlinear Biodynamics in Categories, Higher dimensional Algebra and \L ukasiewicz-Moisil Topos: Transformation of Neural, Genetic and Neoplastic Networks, Axiomathes,16: 65--122(2006).
  7. ↑ Baianu, I.C. and M. Marinescu: 1974, A Functorial Construction of (M,R) -- Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19 : 388-391.
  8. ↑ Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics , 39 : 249-258.
  9. ↑ Baianu, I.C.: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42 : 431-446
  10. ↑ Baianu, I. C.: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), \emph{Mathematical Models in Medicine}, vol. 7., Pergamon Press, New York, 1513-1577; CERN Preprint No. EXT-2004-072
  11. ↑ Baianu, I. C.: 2006, Robert Rosen's Work and Complex Systems Biology, Axiomathes 16 (1--2): 25--34.
  12. ↑ Kuhn, Stephen T., 1983, "An Axiomatization of Predicate Functor Logic.", Notre Dame Journal of Formal Logic 24: 233--41.
  13. ↑ Willard Quine. 1976. "Algebraic Logic and Predicate Functors." in Ways of Paradox and Other Essays , enlarged ed. Harvard Univ. Press: 283--307.
  14. ↑ Willard Quine. 1982. Methods of Logic , 4th ed. Harvard Univ. Press. Chpt. 45.

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