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PlanetPhysics/Category Theory Applications

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Category Theory Applications in Physics edit

A `categorification' of theoretical physics (including quantum field theories) began as early as 1968 [1]; this `categorification' process in physics continues today, especially after 1985 ([2] and references cited therein). Other recent applications are in computer programming theory, automata theory, ontology/categorical ontology [3] and linguistics.

Category Theory Applications in Mathematical Biophysics edit

categories of sets were first introduced in mathematical biophysics in 1958 [4], followed by the introduction of biotheoretical models in categories with structure in 1968-1971 [5]. For a review of the earlier work see ref. [2].

All Sources edit

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [2] [20] [21] [3] [22] [23] [24] [25] [26] [27] [28] [29] [30] [22] [31] [32] [33] [34] [35]

References edit

  1. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Baianu-Marinescu68,Baianu71a,Baianu71b
  2. ↑ 2.0 2.1 2.2 Baianu, I.C.: 1987. Mathematical Models in Medicine , vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577; URLs: CERN Preprint No. EXT-2004-072: , available here as PDF, or as as an archived html document.
  3. ↑ 3.0 3.1 Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
  4. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Rosen58a,Rosen58b
  5. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Baianu-Marinescu68,Baianu70, Baianu71a, Baianu71b
  6. ↑ Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: Advances in Mathematics , 135, 145--206.
  7. ↑ Baez, J. \& Dolan, J., 1998b, ``Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1--36.
  8. ↑ Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited -- 2001 and Beyond , Berlin: Springer, 29--50.
  9. ↑ Baez, J., 1997, An Introduction to n-Categories, in Category Theory and Computer Science, Lecture Notes in Computer Science , 1290, Berlin: Springer-Verlag, 1--33.
  10. ↑ Baianu, I. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. Bulletin of Mathematical Biophysics 30 , 148-159.
  11. ↑ Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics , 32 : 539-561.
  12. ↑ Baianu,I.C. : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Ibid. , 33 (3), 339--354.
  13. ↑ I.C. Baianu: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science , September 1--4, 1971, University of Bucharest.
  14. ↑ I.C. Baianu: \L ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
  15. ↑ I.C. Baianu and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin of Mathematical Biophysics , 35 (4), 475--486.
  16. ↑ I.C. Baianu: 1973, Some Algebraic Properties of (M,R) -- Systems. Bulletin of Mathematical Biophysics 35 , 213-217.
  17. ↑ I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of (M,R) -- Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19 : 388-391.
  18. ↑ I.C. Baianu: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics , 39 : 249-258.
  19. ↑ I.C. Baianu: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42 : 431-446.
  20. ↑ 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, [\\http://www.ag.uiuc.edu/fs401/QAuto.pdf PDF's of Abstract and Preprint of Report].
  21. ↑ 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, Axiomathes 16 Nos. 1--2, 65--122.
  22. ↑ 22.0 22.1 Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43 : 757-831.
  23. ↑ Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 : 231-294.
  24. ↑ Eilenberg, S. \& Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press.
  25. ↑ Grothendieck, A. et al., S\'eminaire de G\'eom\'etrie Alg\'ebrique, Vol. 1--7, Berlin: Springer-Verlag.
  26. ↑ Grothendieck, A., 1957, "Sur Quelques Points d'alg\`ebre homologique", Tohoku Mathematics Journal, 9, 119--221.
  27. ↑ 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.
  28. ↑ Lawvere, F. W., 1965, "Algebraic Theories, Algebraic Categories, and Algebraic Functors", Theory of Models, Amsterdam: North Holland, 413--418.
  29. ↑ 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.
  30. ↑ MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.
  31. ↑ Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 : 231-294.
  32. ↑ 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., Axiomathes , 16 Nos. 1--2: 65--122.
  33. ↑ Rosen, R.: 1958a, A Relational Theory of Biological Systems., Bulletin of Mathematical Biophysics 20 : 245-260.
  34. ↑ Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., Bulletin of Mathematical Biophysics 20 : 317-341.
  35. ↑ See also a more extensive category theory bibliography
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