Talk:PlanetPhysics/Category Theory Applications

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\subsection{Category Theory Applications in Physics}

A `categorification' of \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} (including \htmladdnormallink{quantum field theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html}) began as early as 1968 \cite{Baianu-Marinescu68,Baianu71a,Baianu71b}; this `categorification' process in physics continues today, especially after 1985 (\cite{Baianu87} and references cited therein). Other recent applications are in \htmladdnormallink{computer}{http://planetphysics.us/encyclopedia/SupercomputerArchitercture.html} \htmladdnormallink{programming}{http://planetphysics.us/encyclopedia/ComputerProgram.html} theory, \htmladdnormallink{automata theory}{http://planetphysics.us/encyclopedia/LQG2.html}, ontology/categorical ontology \cite{Bggb4} and linguistics.

\subsection{Category Theory Applications in Mathematical Biophysics} \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} of sets were first introduced in mathematical biophysics in 1958 \cite{Rosen58a,Rosen58b}, followed by the introduction of biotheoretical models in \htmladdnormallink{categories with structure}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in 1968-1971 \cite{Baianu-Marinescu68,Baianu70, Baianu71a, Baianu71b}. For a review of the earlier \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} see ref. \cite{Baianu87}.

\begin{thebibliography}{99}

\bibitem{BAJ-DJ98a} Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: \emph{Advances in Mathematics}, 135, 145--206.

\bibitem{BAJ-DJ98B} Baez, J. \& Dolan, J., 1998b, ``Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1--36.

\bibitem{BAJ-DJ2k1} Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in \emph{Mathematics Unlimited -- 2001 and Beyond}, Berlin: Springer, 29--50.

\bibitem{BAJ-DJ97} Baez, J., 1997, An Introduction to n-Categories, in \emph{Category Theory and Computer Science, Lecture Notes in Computer Science}, 1290, Berlin: Springer-Verlag, 1--33.

\bibitem{Baianu-Marinescu68} Baianu, I. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. \emph{Bulletin of Mathematical Biophysics} \textbf{30}, 148-159.

\bibitem{Baianu70} Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. \emph{Bulletin of Mathematical Biophysics}, \textbf{32}: 539-561.

\bibitem{Baianu71a} Baianu,I.C. : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. \emph{Ibid.}, \textbf{33} (3), 339--354.

\bibitem{Baianu71b} I.C. Baianu: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), \emph{Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science}, September 1--4, 1971, University of Bucharest.

\bibitem{ICB04b} I.C. Baianu: \L ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. \textit{Health Physics and Radiation Effects} (June 29, 2004).

\bibitem{ICBs5} I.C. Baianu and D. Scripcariu: 1973, On Adjoint Dynamical Systems. \emph{The Bulletin of Mathematical Biophysics}, \textbf{35}(4), 475--486.

\bibitem{ICB5} I.C. Baianu: 1973, Some Algebraic Properties of \emph{\textbf{(M,R)}} -- Systems. \emph{Bulletin of Mathematical Biophysics} \textbf{35}, 213-217.

\bibitem{ICBm2} I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of \emph{\textbf{(M,R)}}-- Systems. \emph{Revue Roumaine de Mathematiques Pures et Appliquees} \textbf{19}: 388-391.

\bibitem{ICB6} I.C. Baianu: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biophysics}, \textbf{39}: 249-258.

\bibitem{ICB7} I.C. Baianu: 1980, Natural Transformations of Organismic Structures. \emph{Bulletin of Mathematical Biophysics} \textbf{42}: 431-446.

\bibitem{Baianu87} Baianu, I.C.: 1987. {\em Mathematical Models in Medicine}, vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577; URLs: {\em CERN Preprint No. EXT-2004-072:}, \htmladdnormallink{available here as PDF}{http://doc.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf}, or \htmladdnormallink{as as an archived html document}{http://en.scientificcommons.org/1857371}.

\bibitem{Bgg2} 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, \textbf{(M,R)}--Systems and Their Higher Dimensional Algebra, \htmladdnormallink{PDF's of Abstract and Preprint of Report}{\\http://www.ag.uiuc.edu/fs401/QAuto.pdf}.

\bibitem{BBGG1} Baianu, I.C. Brown R., J. F. Glazebrook, and Georgescu G.: 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: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.

\bibitem{EML1} Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., \emph{American Mathematical Society 43}: 757-831.

\bibitem{EL} Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, \emph{Transactions of the American Mathematical Society} \textbf{58}: 231-294.

\bibitem{ES-CH56} Eilenberg, S. \& Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press.

\bibitem{ALEXsem} Grothendieck, A. et al., S\'eminaire de G\'eom\'etrie Alg\'ebrique, Vol. 1--7, Berlin: Springer-Verlag.

\bibitem{ALEX57} Grothendieck, A., 1957, "Sur Quelques Points d'alg\`ebre homologique", Tohoku Mathematics Journal, 9, 119--221.

\bibitem{LFW64} Lawvere, F. W., 1964, "An Elementary Theory of the Category of Sets", Proceedings of the National Academy of Sciences U.S.A., 52, 1506--1511.

\bibitem{LFW65} Lawvere, F. W., 1965, "Algebraic Theories, Algebraic Categories, and Algebraic Functors", Theory of Models, Amsterdam: North Holland, 413--418.

\bibitem{Lawvere66} Lawvere, F. W., 1966, "The Category of Categories as a Foundation for Mathematics", Proceedings of the Conference on Categorical Algebra, La Jolla, New York: Springer-Verlag, 1--21.

\bibitem{MacLane98} MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.

\bibitem{EML1} Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., \emph{American Mathematical Society 43}: 757-831.

\bibitem{Eilenberg-MacLane45} Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, \emph{Transactions of the American Mathematical Society} \textbf{58}: 231-294.

\bibitem{BBGG2k6} Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., {\em Axiomathes}, \textbf{16} Nos. 1--2: 65--122.

\bibitem{Rosen58a} Rosen, R.: 1958a, A Relational Theory of Biological Systems., {\em Bulletin of Mathematical Biophysics} \textbf{20}: 245-260.

\bibitem{Rosen58b} Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., {\em Bulletin of Mathematical Biophysics} \textbf{20}: 317-341.

\bibitem{Other} See also a more extensive \htmladdnormallink{category theory bibliography}{http://planetphysics.us/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html}

\end{thebibliography}

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