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

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