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%%% Primary Title: categories and supercategories in relational biology
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%%% Filename: CategoriesAndSupercategoriesInRelationalBiology.tex
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\begin{document}
This topic entry introduces one of the most general mathematical models of living organisms
called `\htmladdnormallink{organismic supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html}' (\htmladdnormallink{OS}{http://planetphysics.us/encyclopedia/OS.html}) which can be axiomatically defined to include both
complete self-reproduction of logically defined $\pi$-entities founded in Quine's logic
and \htmladdnormallink{dynamic system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} subject to both \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} and \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} transformations.
\subsection{Organismic Supercategories (OS)}
\emph{OS} mathematical models were introduced as structures in \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} that are mathematical interpretations of the axioms in ETAS- a natural extension of Lawvere's $elementary\; theory\; of \; abstract \; categories$ (\htmladdnormallink{ETAC}{http://planetphysics.us/encyclopedia/ETACAxioms.html}) to \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} structures and heterofunctors.
When regarded as categorical models of supercomplex \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} in living organisms OS provide a unified conceptual framework for \htmladdnormallink{relational biology}{http://planetphysics.us/encyclopedia/RSystemsCategory.html} that utilizes flexible, algebraic and \htmladdnormallink{topological structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} which transform naturally under heteromorphisms or heterofunctors. One of the advantages of the \htmladdnormallink{ETAS}{http://planetphysics.us/encyclopedia/ETACAxioms.html} axiomatic approach, which was inspired by the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} 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 \htmladdnormallink{complex systems biology}{http://planetphysics.us/encyclopedia/SystemsBiology.html} (\cite{Bgg2}, \cite{BBGG1} and references cited therein.)
\subsection{Selected Examples of OS Applications to Relational and Complex Systems Biology}
Whereas super-categories are usually defined as \htmladdnormallink{n-categories}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} 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 \htmladdnormallink{Nicolas Rashevsky}{http://planetphysics.us/encyclopedia/NicolasRashevsky.html}, \htmladdnormallink{Robert Rosen}{http://planetphysics.us/encyclopedia/RobertRosen.html} and by the author, are compared with the classical approach to qualitative dynamics of \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} (QDS). \htmladdnormallink{Natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html} 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 \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} and \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/OS.html} of complex biological systems allow for dynamic \htmladdnormallink{computations}{http://planetphysics.us/encyclopedia/LQG2.html} of cell transformations, neoplasia and cancer.
\begin{thebibliography}{99}
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Bacon, John, 1985, ``The completeness of a predicate--functor logic,'' Journal of Symbolic Logic 50: 903--926.
\bibitem{BP59}
Paul Bernays, 1959, ``Uber eine naturliche Erweiterung des Relationenkalkuls.'' in Heyting, A., ed., Constructivity in Mathematics. North Holland: 1--14.
\bibitem{ICB71}
References [14] to [34] in the ``bibliography of category theory and algebraic topology''
\bibitem{BGB06}
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).
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\bibitem{Bgg2}
Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004,
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\htmladdnormallink{PDF of Abstract and Preprint of Report}{http://fs512.fshn.uiuc.edu/QAuto.pdf}
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Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006,
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Willard Quine. 1982. {\em Methods of Logic}, 4th ed. Harvard Univ. Press. Chpt. 45.
\end{thebibliography}
\end{document}
