Category Theory

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Welcome to the Category Theory International Project !

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  • "In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942-1945, in connection with algebraic topology."


  • This is an International Project on Category Theory, Higher Dimensional Algebra and their Novel Applications, such as:


  • Topoi,
  • n-Categories
  • Nonabelian Algebraic Topology,

Applications and Applied Mathematics

  • Categorical Dynamics,
  • Computational Theory and Logic,
  • Quantum Physics and Quantum Algebraic Topology,
  • Complex Systems and Relational Biology,
    • Mathematical Medicine
    • Ecosystems
    • Biosphere
  • Sociology
  • Categorical Ontology
  • Philosophy of Science
  • (Specify your own novel application field, such as Anabelian Geometry, Noncommutative Geometry, Nondistributive Logics, Monassociative Mathematics, NonNewtonian analysis, etc)


w:Category theory topics w:Homological algebra: w:Abelian categoryw:Sheaf theoryw:K-theory

w:Topos theoryw:Enriched category theoryw:Higher category theory

  w:Monoidal categoryw:Closed categoryw:Dagger category

More category theory topics

Categorical diagram of a pushout



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List of Participants

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  2. 2 User:Bci21
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Bibliography: References

  1. [1] H. CARTAN et C. CHEVALLEY, Séminaire de l'École Normale Supérieure, 8e année (1955-1956), Géométrie algébrique. Numdam | Zbl 0074.36602
  2. [2] H. CARTAN and S. EILENBERG, Homological Algebra, Princeton Math. Series (Princeton University Press), 1956. MR 17,1040e | Zbl 0075.24305
  3. [3] W. L. CHOW and J. IGUSA, Cohomology theory of varieties over rings, Proc. Nat. Acad. Sci. U.S.A., t. XLIV (1958), p. 1244-1248. MR 20 #6427 | Zbl 0096.36002
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  5. [5] H. GRAUERT, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Math. Inst. Hautes Études Scient., n° 5, 1960. Numdam | Zbl 0100.08001
  6. [6] A. GROTHENDIECK, Sur quelques points d'algèbre homologique, Tôhoku Math. Journ., t. IX (1957), p. 119-221. Article | MR 21 #1328 | Zbl 0118.26104
  7. [7] A. GROTHENDIECK, Cohomology theory of abstract algebraic varieties, Proc. Intern. Congress of Math., p. 103-118, Edinburgh (1958). MR 24 #A733 | Zbl 0119.36902
  8. [8] A. GROTHENDIECK, Géométrie formelle et géométrie algébrique, Séminaire Bourbaki, 11e année (1958-1959), exposé 182. Numdam | Zbl 0229.14005
  9. [9] M. NAGATA, A general theory of algebraic geometry over Dedekind domains, Amer. Math. Journ. : I, t. LXXVIII, p. 78-116 (1956) ; II, t. LXXX, p. 382-420 (1958). MR 18,600e | Zbl 0089.26403
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  11. [11] P. SAMUEL, Commutative algebra (Notes by D. Herzig), Cornell Univ., 1953.
  12. [12] P. SAMUEL, Algèbre locale, Mém. Sci. Math., n° 123, Paris, 1953. Numdam | MR 14,1012c | Zbl 0053.01901
  13. [13] P. SAMUEL and O. ZARISKI, Commutative algebra, 2 vol., New York (Van Nostrand), 1958-1960. Zbl 0322.13001
  14. [14] J.-P. SERRE, Faisceaux algébriques cohérents, Ann. of Math., t. LXI (1955), p. 197-278. MR 16,953c | Zbl 0067.16201
  15. [15] J.-P. SERRE, Sur la cohomologie des variétés algébriques, Journ. de Math. (9), t. XXXVI (1957), p. 1-16. MR 18,765b | Zbl 0078.34604
  16. [16] J.-P. SERRE, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, t. VI (1955-1956), p. 1-42. Numdam | MR 18,511a | Zbl 0075.30401
  17. [17] J.-P. SERRE, Sur la dimension homologique des anneaux et des modules noethériens, Proc. Intern. Symp. on Alg. Number theory, p. 176-189, Tokyo-Nikko, 1955. Zbl 0073.26004
  18. [18] A. WEIL, Foundations of algebraic geometry, Amer. Math. Soc. Coll. Publ., n° 29, 1946. MR 9,303c | Zbl 0063.08198
  19. [19] A. WEIL, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., t. LV (1949), p. 497-508. Article | MR 10,592e | Zbl 0032.39402
  20. [20] O. ZARISKI, Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc., n° 5 (1951). MR 12,853f | Zbl 0045.24001
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  22. [22] E. KÄHLER, Geometria Arithmetica, Ann. di Mat. (4), t. XLV (1958), p. 1-368. Zbl 0142.18101

Topical references for Categories and Algebraic Topology Applications in Theoretical Physics

  1. 1,Adámek, J.. et al., Locally Presentable and Accessible Categories, Cambridge: Cambridge University Press (1994).
  2. 2 Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkh'auser, Boston-Basel-Berlin (2003).
  3. 3 Atiyah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France, 84: 307-317.
  4. 3. Auslander, M. 1965. Coherent Functors. Proc. Conf. Cat. Algebra, La Jolla, 189-231.
  5. Awodey, S. & Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168-1182.
  6. 5 Awodey, S. & Reck, E. R., 2002, Completeness and Categoricity I. Nineteen-Century Axiomatics to Twentieth-Century Metalogic., History and Philosophy of Logic, 23, 1, 1-30.
  7. Awodey, S. & Reck, E. R., 2002, Completeness and Categoricity II. Twentieth-Century Metalogic to Twenty-first-Century Semantics, History and Philosophy of Logic, 23, 2, 77-94.
  8. 6 Awodey, S., 1996, Structure in Mathematics and Logic: A Categorical Perspective, Philosophia Mathematica, 3, 209-237.
  9. 7 Awodey, S., 2004, An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism., Philosophia Mathematica, 12, 54-64.
  10. 8 Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
  11. 9 Baez, J. and Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes., Advances in Mathematics, 135, 145-206.

10 Baez, J. and Dolan, J., 1998b, ``Categorification, Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1-36. 11 Baez, J. and Dolan, J., 2001, ``From Finite Sets to Feynman Diagrams, Mathematics Unlimited - 2001 and Beyond, Berlin: Springer, 29-50. 12 Baez, J., 1997, ``An Introduction to n-Categories, Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1-33.

  1. 13 Baianu, I.C. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. Bulletin of Mathematical Biophysics 30, 148-159.

14 Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561. 15 Baianu, I.C.: 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Ibid., 33 (3), 339-354. 15 Baianu, I.C.: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science, September 1-4, 1971, Bucharest. 16 Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. Bulletin of Mathematical Biophysics, 35(4), 475-486. 17 Baianu, I.C.: 1973, Some Algebraic Properties of (M,R) - Systems. Bulletin of Mathematical Biophysics 35, 213-217. 18 Baianu, I.C. and M. Marinescu: 1974, On A Functorial Construction of (M,R)- Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391. 19 Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258. 20 Baianu, I.C.: 1980a, Natural Transformations of Organismic Structures., Bulletin of Mathematical Biology,42: 431-446. 20 Baianu, I. C.: 1983, Natural Transformation Models in Molecular Biology., in Proceedings of the SIAM Natl. Meet., Denver,CO.; Eprint at cogprints.org/3675 20 Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic Networks, FASEB Proceedings 43, 917. 21 Baianu, I. C.: 1986-1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577; URLs: CERN Preprint No. EXT-2004-072 , and html Abstract. 22 Baianu, I. C.: 1987b, Molecular Models of Genetic and Organismic Structures, in Proceed. Relational Biology Symp. Argentina; CERN Preprint No.EXT-2004-067 . 23 Baianu, I.C.: 2004a. Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint: w. Cogprints at Sussex Univ. 24 Baianu, I.C.: 2004b Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN EXT-2004-059,Health Physics and Radiation Effects , (June 29, 2004). 25 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, Abstract and Preprint of Report. 26 Baianu, I.C.: 2004a, Quantum Nano-Automata (QNA): Microphysical Measurements with Microphysical QNA Instruments, CERN Preprint EXT-2004-125. 27 Baianu, I. C.: 2004b, Quantum Interactomics and Cancer Mechanisms, Preprint 00001978 . 28 Baianu, I. C.: 2006, Robert Rosen's Work and Complex Systems Biology, Axiomathes 16(1-2):25-34. 29 Baianu, I. C., Brown, R. and J. F. Glazebrook: 2006, Quantum Algebraic Topology and Field Theories. Preprint 30 Baianu, I.C.: 2008, Translational Genomics and Human Cancer Interactomics, (invited Review, submitted in November 2007 to Translational Oncogenomics).

  1. 31 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.
  2. 32 Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007a, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, 17: 35-168.
  3. 33 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. 34 Baianu, I.C. et al. Quantum Algebra and Symmetries. PediaPress:Mainz, Germany, 1,112 pages, volumes I-III, Second edition. Books: ``Quantum Algebra and Symmetries
  5. 35 M. Barr and C. Wells. Toposes, Triples and Theories. Montreal: McGill University, 2000.
  6. 36 Barr, M. & Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
  7. 37 Barr, M. & Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
  8. 38 Batanin, M., 1998, Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-Categories", Advances in Mathematics, 136, 39-103.
  9. 39 Bell, J. L., 1981, Category Theory and the Foundations of Mathematics, British Journal for the Philosophy of Science, 32, 349-358.
  10. 40 Bell, J. L., 1982, Categories, Toposes and Sets, Synthese,51, 3, 293-337.

41 Bell, J. L., 1986, From Absolute to Local Mathematics, Synthese, 69, 3, 409-426. 42 Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press. 43 Birkoff, G. and Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS. 44 Biss, D.K., 2003, Which Functor is the Projective Line?, American Mathematical Monthly, 110, 7, 574-592. 45 Blass, A. and Scedrov, A., 1983, Classifying Topoi and Finite Forcing , Journal of Pure and Applied Algebra, 28, 111-140. 46 Blass, A. and Scedrov, A., 1989, Freyd's Model for the Independence of the Axiom of Choice, Providence: AMS. 47 Blass, A. and Scedrov, A., 1992, Complete Topoi Representing Models of Set Theory, Annals of Pure and Applied Logic , 57, no. 1, 1-26. 48 Blass, A., 1984, The Interaction Between Category Theory and Set Theory., Mathematical Applications of Category Theory, 30, Providence: AMS, 5-29. 49 Blute, R. & Scott, P., 2004, Category Theory for Linear Logicians., in Linear Logic in Computer Science 50 Borceux, F.: 1994, Handbook of Categorical Algebra, vols: 1-3, in Encyclopedia of Mathematics and its Applications 50 to 52, Cambridge University Press. 51 Bourbaki, N. 1961 and 1964: Algèbre commutative., in Éléments de Mathématique., Chs. 1-6., Hermann: Paris. 52 R. Brown: Topology and Groupoids, BookSurge LLC (2006). 53 Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, Applied Categorical Structures 12: 63-80. 54 Brown, R., Higgins, P. J. and R. Sivera,: 2007a, Non-Abelian Algebraic Topology, in preparation. http://www.bangor.ac.uk/ mas010/nonab-a-t.html ; http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf 55 Brown, R., Glazebrook, J. F. and I.C. Baianu.: 2007b, A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly Complex Structures and Dynamics., Axiomathes (17): 321-379. 56 Brown, R., Paton, R. and T. Porter.: 2004, Categorical language and hierarchical models for cell systems, in Computation in Cells and Tissues - Perspectives and Tools of Thought, Paton, R.; Bolouri, H.; Holcombe, M.; Parish, J.H.; Tateson, R. (Eds.) Natural Computing Series, Springer Verlag, 289-303. 57 Brown R. and T. Porter: 2003, Category theory and higher dimensional algebra: potential descriptive tools in neuroscience, In: Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1, 80-92. 58 Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., Theory and Applications of Categories 10, 71-93. 59 Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr., 71: 273-286. 60 Brown, R. and T. Porter: 2006, Category Theory: an abstract setting for analogy and comparison, In: What is Category Theory?, Advanced Studies in Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. 61 Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. Géom. Diff. 17, 343-362. 62 Brown R, and Porter T (2006) Category theory: an abstract setting for analogy and comparison. In: What is category theory? Advanced studies in mathematics and logic. Polimetrica Publisher, Italy, pp. 257-274. 63 Brown R, Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. LMS J. Comput. Math., 2: 25-61. 64 Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80: 1-34. 64 Buchsbaum, D. A.: 1969, A note on homology in categories., Ann. of Math. 69: 66-74. 65 Bucur, I. (1965). Homological Algebra. (orig. title: ``Algebra Omologica) Ed. Didactica si Pedagogica: Bucharest. 66 Bucur, I., and Deleanu A. (1968). Introduction to the Theory of Categories and Functors. J.Wiley and Sons: London 67 Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179, 291-317. 68 Bunge, M., 1974, "Topos Theory and Souslin's Hypothesis", Journal of Pure and Applied Algebra, 4, 159-187. 69 Bunge, M., 1984, "Toposes in Logic and Logic in Toposes", Topoi, 3, no. 1, 13-22. 70 Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math, 179: 291-317. 71 Butterfield J., Isham C.J. (2001) Spacetime and the philosophical challenges of quantum gravity. In: Callender C, Hugget N (eds) Physics meets philosophy at the Planck scale. Cambridge University Press, pp 33-89. 72 Butterfield J., Isham C.J. 1998, 1999, 2000-2002, A topos perspective on the Kochen-Specker theorem I-IV, Int J Theor Phys 37(11):2669-2733; 38(3):827-859; 39(6):1413-1436; 41(4): 613-639. 73 Cartan, H. and Eilenberg, S. 1956. Homological Algebra, Princeton Univ. Press: Pinceton. 74 M. Chaician and A. Demichev. 1996. Introduction to Quantum Groups, World Scientific . 75 Chevalley, C. 1946. The theory of Lie groups. 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Géom. Diff. Glob. Bruxelles, pp.137-150. 93 Ehresmann, C.:1963, Catégories doubles des quintettes: applications covariantes , C.R.A.S. Paris, 256: 1891-1894. 94 Ehresmann, A. C. & Vanbremeersch, J-P., 1987, "Hierarchical Evolutive Systems: a Mathematical Model for Complex Systems", Bulletin of Mathematical Biology, 49, no. 1, 13-50. 95 Ehresmann, C.: 1984, Oeuvres complètes et commentées: Amiens, 1980-84, edited and commented by Andrée Ehresmann. 96 Ehresmann, A. C. and J.-P. Vanbremersch: 1987, Hierarchical Evolutive Systems: A mathematical model for complex systems, Bull. of Math. Biol. 49 (1): 13-50. 97 Ehresmann, A. C. and J.-P. Vanbremersch: 2006, The Memory Evolutive Systems as a model of Rosen's Organisms, Axiomathes 16 (1-2): 13-50. 98 Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831. 99 Eilenberg, S. and S. 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(USA), Volume 29, Issue 5, pp. 155-158. 107 Ellerman, D., 1988, "Category Theory and Concrete Universals", Synthese, 28, 409-429. 108 Z. F. Ezawa, G. Tsitsishvilli and K. Hasebe : Noncommutative geometry, extended algebra and Grassmannian solitons in multicomponent Hall systems, arXiv:hep-th/0209198. 109 Feferman, S., 1977, ``Categorical Foundations and Foundations of Category Theory, Logic, Foundations of Mathematics and Computability, R. Butts (ed.), Reidel, 149-169. 110 Fell, J. M. G., 1960. ``The Dual Spaces of C*-Algebras, Transactions of the American Mathematical Society, 94: 365-403. 111 Feynman, R. P., 1948, ``A Space-Time Approach to Non-Relativistic Quantum Mechanics., Reviews of Modern Physics, 20: 367--387. [It is reprinted in (Schwinger 1958).] 112 Freyd, P., 1960. Functor Theory (Dissertation). Princeton University, Princeton, New Jersey. 113 Freyd, P., 1963, Relative homological algebra made absolute. , Proc. Natl. Acad. USA, 49:19-20. 114 Freyd, P., 1964, Abelian Categories. An Introduction to the Theory of Functors, New York and London: Harper and Row. 115 Freyd, P., 1965, The Theories of Functors and Models., Theories of Models, Amsterdam: North Holland, 107-120. 116 Freyd, P., 1966, Algebra-valued Functors in general categories and tensor product in particular., Colloq. 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