category theory applications

Category Theory Applications in Physics

A `categorification' of theoretical physics (including quantum field theories) began as early as 1968 [5,7,8]; this `categorification' process in physics continues today, especially after 1985 ([15] and references cited therein). Other recent applications are in computer programming theory, automata theory, ontology/categorical ontology [18] and linguistics.

Category Theory Applications in Mathematical Biophysics

categories of sets were first introduced in mathematical biophysics in 1958 [31,32], followed by the introduction of biotheoretical models in categories with structure in 1968-1971 [5,6,7,8]. For a review of the earlier work see ref. [15].

Bibliography

1
Baez, J. & Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: Advances in Mathematics, 135, 145-206.

2
Baez, J. & Dolan, J., 1998b, ``Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1-36.

3
Baez, J. & Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited - 2001 and Beyond, Berlin: Springer, 29-50.

4
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.

5
Baianu, I. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. Bulletin of Mathematical Biophysics 30, 148-159.

6
Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.

7
Baianu,I.C. : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Ibid., 33 (3), 339-354.

8
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.

9
I.C. Baianu: Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).

10
I.C. Baianu and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin of Mathematical Biophysics, 35(4), 475-486.

11
I.C. Baianu: 1973, Some Algebraic Properties of (M,R) - Systems. Bulletin of Mathematical Biophysics 35, 213-217.

12
I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of (M,R)- Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.

13
I.C. Baianu: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.

14
I.C. Baianu: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42: 431-446.

15
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.

16
Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)-Systems and Their Higher Dimensional Algebra, PDF's of Abstract and Preprint of Report.

17
Baianu, I.C. Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic networks, Axiomathes 16 Nos. 1-2, 65-122.

18
Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.

28
Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831.

20
Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.

21
Eilenberg, S. & Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press.

22
Grothendieck, A. et al., Séminaire de Géométrie Algébrique, Vol. 1-7, Berlin: Springer-Verlag.

23
Grothendieck, A., 1957, "Sur Quelques Points d'algèbre homologique", Tohoku Mathematics Journal, 9, 119-221.

24
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.

25
Lawvere, F. W., 1965, "Algebraic Theories, Algebraic Categories, and Algebraic Functors", Theory of Models, Amsterdam: North Holland, 413-418.

26
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.

27
MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.

28
Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831.

29
Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.

30
Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1-2: 65-122.

31
Rosen, R.: 1958a, A Relational Theory of Biological Systems., Bulletin of Mathematical Biophysics 20: 245-260.

32
Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., Bulletin of Mathematical Biophysics 20: 317-341.

33
See also a more extensive category theory bibliography



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