Wikiversity

PlanetPhysics/Axiomatics and Categorical Foundations of Mathematical Physics

< PlanetPhysics

Axiomatics and Categorical Foundations of Mathematical Physics

edit

This is a contributed new topic on the mathematical foundations of theoretical physics and quantum theories.

  1. Axiomatic foundations of quantum field theories
  2. quantum logics and logic algebras: Post and logics
  3. Index of Quantum Algebraic Topology #adjointness, equivalence, isomorphism at the foundations of categorical physics #category theory in quantum physics and general relativity #Categories of quantum logic algebras
  4. functor categories and super-categories #index of category theory #indexes of category
  5. classification of -algebras and groupoid convolution -algebras
  6. Quantum topoi and quantum logic extended-toposes
  7. non-Abelian structures and gauge theories
  8. Non-Abelian Quantum Algebraic Topology and AQFT #Classical and categorical Galois theories of quantum groups and quantum groupoids
  9. Theory of quantum computation: quantum logics, quantum automata and quantum computation
  10. Measure theory and probability in quantum statistical mechanics #quantum symmetries and quantum groupoid representation theory
  11. noncommutative geometry, SUSY and axiomatic quantum gravity (AQG)

Literature references for mathematical physics foundations: axiomatics and categories

All Sources

edit

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] [187] [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255] [256] [257] [258] [259]

References

edit
  1. ↑ Alfsen, E.M. and F. W. Schultz: \emph{Geometry of State Spaces of Operator Algebras}, Birkhauser, Boston--Basel--Berlin (2003).
  2. ↑ Atiyah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France , 84 : 307--317.
  3. ↑ Awodey, S. \& Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168--1182.
  4. ↑ Awodey, S., 1996, Structure in Mathematics and Logic: A Categorical Perspective, Philosophia Mathematica , 3: 209--237.
  5. ↑ Awodey, S., 2004, An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism, Philosophia Mathematica , 12: 54--64.
  6. ↑ Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
  7. ↑ Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: Advances in Mathematics , 135, 145--206.
  8. ↑ Baez, J. \& Dolan, J., 1998b, "Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1--36.
  9. ↑ Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited -- 2001 and Beyond , Berlin: Springer, 29--50.
  10. ↑ 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.
  11. ↑ 11.0 11.1 Baianu, I.C.: 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Ibid. , 33 (3), 339--354. Cite error: Invalid <ref> tag; name "ICB4" defined multiple times with different content
  12. ↑ Baianu, I.C.: 2004a, Quantum Nano--Automata (QNA): Microphysical Measurements with Microphysical QNA Instruments, CERN Preprint EXT--2004--125 .
  13. ↑ 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.
  14. ↑ Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes , 17: 169-225.
  15. ↑ Barr, M and C.~Wells. Toposes, Triples and Theories . Montreal: McGill University, 2000.
  16. ↑ Barr, M. \& Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
  17. ↑ Batanin, M., 1998, Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-Categories, Advances in Mathematics , 136: 39--103.
  18. ↑ Bell, J. L., 1981, Category Theory and the Foundations of Mathematics, British Journal for the Philosophy of Science , 32, 349--358.
  19. ↑ Bell, J. L., 1982, Categories, Toposes and Sets, Synthese , 51, 3, 293--337.
  20. ↑ Bell, J. L., 1986, From Absolute to Local Mathematics, Synthese , 69, 3, 409--426.
  21. ↑ Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
  22. ↑ Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
  23. ↑ Biss, D.K., 2003, Which Functor is the Projective Line?, American Mathematical Monthly , 110, 7, 574--592.
  24. ↑ Blass, A. \& Scedrov, A., 1983, Classifying Topoi and Finite Forcing , Journal of Pure and Applied Algebra, 28, 111--140.
  25. ↑ Blass, A., 1984, The Interaction Between Category Theory and Set Theory., Mathematical Applications of Category Theory, 30, Providence: AMS, 5--29.
  26. ↑ Borceux, F.: 1994, Handbook of Categorical Algebra , vols: 1--3, in Encyclopedia of Mathematics and its Applications 50 to 52 , Cambridge University Press.
  27. ↑ Bourbaki, N. 1961 and 1964: Alg\`{e bre commutative.}, in \`{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris.
  28. ↑ R. Brown: Topology and Groupoids , BookSurge LLC (2006).
  29. ↑ 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.
  30. ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2007a, Non-Abelian Algebraic Topology , (vol. 2 in preparation; vol. 1 is available at Bangor Univ. R. Brown's website for download as PDF.)
  31. ↑ 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.
  32. ↑ 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.
  33. ↑ Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr. , 71: 273-286.
  34. ↑ Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. G\'{e om. Diff.} 17 , 343-362.
  35. ↑ 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.
  36. ↑ 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.
  37. ↑ Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80 : 1-34.
  38. ↑ Bucur, I., and Deleanu A. (1968). Introduction to the Theory of Categories and Functors . J.Wiley and Sons: London
  39. ↑ Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179 , 291-317.
  40. ↑ Bunge, M., 1974, Topos Theory and Souslin's Hypothesis, Journal of Pure and Applied Algebra, 4, 159-187.
  41. ↑ Bunge, M., 1984, Toposes in Logic and Logic in Toposes, Topoi, 3, no. 1, 13-22.
  42. ↑ Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math , \textbf {179}: 291-317.
  43. ↑ 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.
  44. ↑ 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.
  45. ↑ Cartan, H. and Eilenberg, S. 1956. Homological Algebra , Princeton Univ. Press: Pinceton.
  46. ↑ M. Chaician and A. Demichev. 1996. Introduction to Quantum Groups, World Scientific .
  47. ↑ Chevalley, C. 1946. The theory of Lie groups. Princeton University Press, Princeton NJ
  48. ↑ Cohen, P.M. 1965. Universal Algebra , Harper and Row: New York, london and Tokyo.
  49. ↑ M. Crainic and R. Fernandes.2003. Integrability of Lie brackets, Ann.of Math . 157 : 575-620.
  50. ↑ Connes A 1994. Noncommutative geometry . Academic Press: New York.
  51. ↑ Croisot, R. and Lesieur, L. 1963. Alg\`ebre noeth\'erienne non-commutative. , Gauthier-Villard: Paris.
  52. ↑ Crole, R.L., 1994, Categories for Types , Cambridge: Cambridge University Press.
  53. ↑ Dieudonn\'e, J. \& Grothendieck, A., 1960, [1971], \'El\'ements de G\'eom\'etrie Alg\'ebrique, Berlin: Springer-Verlag.
  54. ↑ Dirac, P. A. M., 1930, The Principles of Quantum Mechanics , Oxford: Clarendon Press.
  55. ↑ Dirac, P. A. M., 1933, The Lagrangian in Quantum Mechanics , Physikalische Zeitschrift der Sowietunion, 3 : 64-72.
  56. ↑ Dirac, P. A. M.,, 1943, Quantum Electrodynamics , Communications of the Dublin Institute for Advanced Studies, A1 : 1-36.
  57. ↑ Dixmier, J., 1981, Von Neumann Algebras, Amsterdam: North-Holland Publishing Company. [First published in French in 1957: Les Algebres d'Operateurs dans l'Espace Hilbertien, Paris: Gauthier--Villars.]
  58. ↑ M. Durdevich : Geometry of quantum principal bundles I, Commun. Math. Phys. 175 (3) (1996), 457--521.
  59. ↑ M. Durdevich : Geometry of quantum principal bundles II, Rev. Math. Phys. 9 (5) (1997), 531--607.
  60. ↑ 60.0 60.1 Ehresmann, C.: 1965, Cat\'egories et Structures , Dunod, Paris. Cite error: Invalid <ref> tag; name "EC" defined multiple times with different content
  61. ↑ Ehresmann, C.: 1952, Structures locales et structures infinit\'esimales, C.R.A.S. Paris 274 : 587-589.
  62. ↑ Ehresmann, C.: 1959, Cat\'egories topologiques et cat\'egories diff\'erentiables, Coll. G\'eom. Diff. Glob. Bruxelles, pp.137-150.
  63. ↑ Ehresmann, C.:1963, Cat\'egories doubles des quintettes: applications covariantes , C.R.A.S. Paris , 256 : 1891--1894.
  64. ↑ Ehresmann, C.: 1984, Oeuvres compl\`etes et comment\'ees: Amiens, 1980-84 , edited and commented by Andr\'ee Ehresmann.
  65. ↑ Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43 : 757-831.
  66. ↑ Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 : 231-294.
  67. ↑ Eilenberg, S. \& MacLane, S., 1942, Group Extensions and Homology, Annals of Mathematics, 43, 757--831.
  68. ↑ Eilenberg, S. \& Steenrod, N., 1952, Foundations of Algebraic Topology, Princeton: Princeton University Press.
  69. ↑ Eilenberg, S.: 1960. Abstract description of some basic functors., J. Indian Math.Soc., 24 :221-234.
  70. ↑ S.Eilenberg. Relations between Homology and Homotopy Groups. Proc.Natl.Acad.Sci.USA (1966),v:10--14.
  71. ↑ Ellerman, D., 1988, Category Theory and Concrete Universals, Synthese, 28, 409--429.
  72. ↑ Z. F. Ezawa, G. Tsitsishvilli and K. Hasebe : Noncommutative geometry, extended   algebra and Grassmannian solitons in multicomponent Hall systems, arXiv:hep--th/0209198.
  73. ↑ Fell, J. M. G., 1960. The Dual Spaces of C*-Algebras, Transactions of the American Mathematical Society , 94: 365-403.
  74. ↑ Feynman, R. P., 1948, Space--Time Approach to Non--Relativistic Quantum Mechanics., Reviews of Modern Physics , 20: 367--387. [It is reprinted in (Schwinger 1958).]
  75. ↑ Freyd, P., 1960. Functor Theory (Dissertation). Princeton University, Princeton, New Jersey.
  76. ↑ Freyd, P., 1963, Relative homological algebra made absolute. , Proc. Natl. Acad. USA , 49 :19-20.
  77. ↑ Freyd, P., 1964, Abelian Categories. An Introduction to the Theory of Functors, New York and London: Harper and Row.
  78. ↑ Freyd, P., 1965, The Theories of Functors and Models., Theories of Models , Amsterdam: North Holland, 107--120.
  79. ↑ Freyd, P., 1966, Algebra-valued Functors in general categories and tensor product in particular., Colloq. Mat . {14}: 89--105.
  80. ↑ Freyd, P., 1972, Aspects of Topoi,Bulletin of the Australian Mathematical Society , 7 : 1--76.
  81. ↑ Freyd, P., 1990, Categories, Allegories, Amsterdam: North Holland.
  82. ↑ Freyd, P., 2002, "Cartesian Logic", Theoretical Computer Science, 278, no. 1--2, 3--21.
  83. ↑ Freyd, P., Friedman, H. \& Scedrov, A., 1987, "Lindembaum Algebras of Intuitionistic Theories and Free Categories"., Annals of Pure and Applied Logic, 35, 2, 167--172.
  84. ↑ Gablot, R. 1971. Sur deux classes de cat\'{e}gories de Grothendieck. Thesis.. Univ. de Lille.
  85. ↑ Gabriel, P.: 1962, Des cat\'egories ab\'eliennes, \emph{Bull. Soc. Math. France} 90 : 323-448.
  86. ↑ Gabriel, P. and M.Zisman:. 1967: Category of fractions and homotopy theory , Ergebnesse der math. Springer: Berlin.
  87. ↑ Gabriel, P. and N. Popescu: 1964, Caract\'{e}risation des cat\'egories ab\'eliennes avec g\'{e}n\'{e}rateurs et limites inductives. , CRAS Paris 258 : 4188-4191.
  88. ↑ Galli, A. \& Reyes, G. \& Sagastume, M., 2000, Completeness Theorems via the Double Dual Functor, Studia Logical, 64, no. 1, 61--81.
  89. ↑ Gelfan'd, I. and Naimark, M., 1943. On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space., Recueil Math\'ematique [Matematicheskii Sbornik] Nouvelle S\'erie, 12 [54]: 197-213. [Reprinted in C*--algebras: 1943--1993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.]
  90. ↑ Georgescu, G. and C. Vraciu 1970. On the Characterization of \L{}ukasiewicz Algebras. J Algebra , 16 (4), 486-495.
  91. ↑ Ghilardi, S. \& Zawadowski, M., 2002, Sheaves, Games \& Model Completions: A Categorical Approach to Nonclassical Porpositional Logics, Dordrecht: Kluwer.
  92. ↑ Ghilardi, S., 1989, Presheaf Semantics and Independence Results for some Non-classical first-order logics, Archive for Mathematical Logic, 29, no. 2, 125--136.
  93. ↑ Goblot, R., 1968, Cat\'egories modulaires , C. R. Acad. Sci. Paris, S\'erie A. , 267 : 381--383.
  94. ↑ Goblot, R., 1971, Sur deux classes de cat\'egories de Grothendieck, Th\`ese. , Univ. Lille, 1971.
  95. ↑ Goldblatt, R., 1979, Topoi: The Categorical Analysis of Logic, Studies in logic and the foundations of mathematics, Amsterdam: Elsevier North-Holland Publ. Comp.
  96. ↑ Goldie, A. W., 1964, Localization in non-commutative noetherian rings, J.Algebra , 1 : 286-297.
  97. ↑ Godement,R. 1958. Th\'{e}orie des faisceaux. Hermann: Paris.
  98. ↑ Gray, C. W.: 1965. Sheaves with values in a category.,\emph {Topology}, 3: 1-18.
  99. ↑ Grothendieck, A.: 1971, Rev\^{e}tements \'Etales et Groupe Fondamental (SGA1), chapter VI: Cat\'egories fibr\'ees et descente, Lecture Notes in Math. 224 , Springer--Verlag: Berlin.
  100. ↑ Grothendieck, A.: 1957, Sur quelque point d-alg\'{e}bre homologique. , Tohoku Math. J. , 9: 119-121.
  101. ↑ Grothendieck, A. and J. Dieudon\'{e}.: 1960, El\'{e}ments de geometrie alg\'{e}brique., Publ. Inst. des Hautes Etudes de Science , 4 .
  102. ↑ Grothendieck, A. et al., S\'eminaire de G\'eom\'etrie Alg\'ebrique, Vol. 1--7, Berlin: Springer-Verlag.
  103. ↑ Groups Authors: J. Faria Martins, Timothy Porter., On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical  .
  104. ↑ Gruson, L, 1966, Compl\'etion ab\'elienne. Bull. Math.Soc. France , 90 : 17-40.
  105. ↑ K.A. Hardie, K.H. Kamps and R.W. Kieboom. 2000. A homotopy 2-groupoid of a Hausdorff space, Applied Cat. Structures 8: 209--234.
  106. ↑ Hatcher, W. S. 1982. The Logical Foundations of Mathematics , Oxford: Pergamon Press.
  107. ↑ Heller, A. :1958, Homological algebra in Abelian categories., Ann. of Math. 68 : 484-525.
  108. ↑ Heller, A. and K. A. Rowe.:1962, On the category of sheaves., Amer J. Math. 84 : 205-216.
  109. ↑ Hermida, C. \& Makkai, M. \& Power, J., 2000, On Weak Higher-dimensional Categories. I, Journal of Pure and Applied Algebra, 154, no. 1-3, 221--246.
  110. ↑ Hermida, C. \& Makkai, M. \& Power, J., 2001, On Weak Higher-dimensional Categories. II, Journal of Pure and Applied Algebra, 157, no. 2-3, 247--277.
  111. ↑ Hermida, C. \& Makkai, M. \& Power, J., 2002, On Weak Higher-dimensional Categories. III, Journal of Pure and Applied Algebra, 166, no. 1-2, 83--104.
  112. ↑ Higgins, P. J.: 2005, Categories and groupoids , Van Nostrand Mathematical Studies: 32, (1971); \emph{Reprints in Theory and Applications of Categories}, No. 7: 1-195.
  113. ↑ Higgins, Philip J. Thin elements and commutative shells in cubical  -categories. Theory Appl. Categ. 14 (2005), No. 4, 60--74 (electronic). msc: 18D05.
  114. ↑ Hyland, J.M.E. \& Robinson, E.P. \& Rosolini, G., 1990, The Discrete Objects in the Effective Topos, Proceedings of the London Mathematical Society (3), 60, no. 1, 1--36.
  115. ↑ Hyland, J. M..E., 1988, A Small Complete Category, Annals of Pure and Applied Logic, 40, no. 2, 135--165.
  116. ↑ Hyland, J. M .E., 1991, First Steps in Synthetic Domain Theory, Category Theory (Como 1990), Lecture Notes in Mathematics, 1488, Berlin: Springer, 131-156.
  117. ↑ Hyland, J. M.E., 2002, Proof Theory in the Abstract, Annals of Pure and Applied Logic, 114, no. 1--3, 43--78.
  118. ↑ E.Hurewicz. CW Complexes.Trans AMS.1955.
  119. ↑ Ionescu, Th., R. Parvan and I. Baianu, 1970, C. R. Acad. Sci. Paris, S\'erie A. , 269 : 112-116, communiqu\'ee par Louis N\'eel .
  120. ↑ C. J. Isham : A new approach to quantising space--time: I. quantising on a general category, Adv. Theor. Math. Phys. 7 (2003), 331--367.
  121. ↑ Jacobs, B., 1999, Categorical Logic and Type Theory, Amsterdam: North Holland.
  122. ↑ Johnstone, P. T., 1977, Topos Theory, New York: Academic Press.
  123. ↑ Johnstone, P. T., 1979a, Conditions Related to De Morgan's Law, Applications of Sheaves, Lecture Notes in Mathematics, 753, Berlin: Springer, 479--491.
  124. ↑ Johnstone, P. T., 1982, Stone Spaces, Cambridge:Cambridge University Press.
  125. ↑ Johnstone, P. T., 1985, "How General is a Generalized Space?", Aspects of Topology, Cambridge: Cambridge University Press, 77--111.
  126. ↑ Joyal, A. \& Moerdijk, I., 1995, Algebraic Set Theory, Cambridge: Cambridge University Press.
  127. ↑ Van Kampen, E. H.: 1933, On the Connection Between the Fundamental Groups of some Related Spaces, Amer. J. Math. 55 : 261-267
  128. ↑ Kan, D. M., 1958, Adjoint Functors, Transactions of the American Mathematical Society , 87, 294-329.
  129. ↑ Kleisli, H.: 1962, Homotopy theory in Abelian categories.,Can. J. Math. , 14 : 139-169.
  130. ↑ Knight, J.T., 1970, On epimorphisms of non-commutative rings., Proc. Cambridge Phil. Soc. , 25 : 266-271.
  131. ↑ Kock, A., 1981, Synthetic Differential Geometry, London Mathematical Society Lecture Note Series, 51, Cambridge: Cambridge University Press.
  132. ↑ S. Kobayashi and K. Nomizu. Foundations of Differential Geometry Vol I., Wiley Interscience, New York--London 1963.
  133. ↑ H. Krips.1999. Measurement in Quantum Theory. , \emph{The Stanford Encyclopedia of Philosophy} ({Winter 1999 Edition}), Edward N. Zalta (ed.)
  134. ↑ Lam, T. Y., 1966, "The category of noetherian modules.", Proc. Natl. Acad. Sci. USA , 55 : 1038-104.
  135. ↑ Lambek, J. \& Scott, P. J., 1981, Intuitionistic Type Theory and Foundations, Journal of Philosophical Logic, 10, 1, 101--115.
  136. ↑ Lambek, J., 1968, Deductive Systems and Categories I. Syntactic Calculus and Residuated Categories, Mathematical Systems Theory, 2, 287--318.
  137. ↑ Lambek, J., 1969, Deductive Systems and Categories II. Standard Constructions and Closed Categories, Category Theory, Homology Theory and their Applications I, Berlin: Springer, 76--122.
  138. ↑ Lambek, J., 1972, Deductive Systems and Categories III. Cartesian Closed Categories, Intuitionistic Propositional Calculus, and Combinatory Logic, Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, 274, Berlin: Springer, 57--82.
  139. ↑ Lambek, J., 1989A, On Some Connections Between Logic and Category Theory, Studia Logica, 48, 3, 269--278.
  140. ↑ Lambek, J., 1989B, On the Sheaf of Possible Worlds, Categorical Topology and its relation to Analysis, Algebra and Combinatorics, Teaneck: World Scientific Publishing, 36--53.
  141. ↑ Lambek, J., 1994a, Some Aspects of Categorical Logic, in Logic, Methodology and Philosophy of Science IX, Studies in Logic and the Foundations of Mathematics 134, Amsterdam: North Holland, 69--89.
  142. ↑ Lambek, J., 1994b, What is a Deductive System?, What is a Logical System?, Studies in Logic and Computation, 4, Oxford: Oxford University Press, 141--159.
  143. ↑ Lambek, J., 2004, What is the world of Mathematics? Provinces of Logic Determined, Annals of Pure and Applied Logic, 126(1-3), 149--158.
  144. ↑ Lambek, J. and P.~J.~Scott. Introduction to higher order categorical logic . Cambridge University Press, 1986.
  145. ↑ E. C. Lance : Hilbert C*--Modules. \emph{London Math. Soc. Lect. Notes} 210 , Cambridge Univ. Press. 1995.
  146. ↑ Landry, E. \& Marquis, J.-P., 2005, Categories in Context: Historical, Foundational and philosophical, Philosophia Mathematica, 13, 1--43.
  147. ↑ Landry, E., 2001, Logicism, Structuralism and Objectivity, Topoi, 20, 1, 79--95.
  148. ↑ Landsman, N. P.: 1998, Mathematical Topics between Classical and Quantum Mechanics , Springer Verlag: New York.
  149. ↑ N. P. Landsman : Mathematical topics between classical and quantum mechanics. Springer Verlag , New York, 1998.
  150. ↑ N. P. Landsman : Compact quantum groupoids,  
  151. ↑ La Palme Reyes, M., et. al., 1994, The non-Boolean Logic of Natural Language Negation, Philosophia Mathematica, 2, no. 1, 45--68.
  152. ↑ La Palme Reyes, M., et. al., 1999, Count Nouns, Mass Nouns, and their Transformations: a Unified Category-theoretic Semantics, in Language, Logic and Concepts , Cambridge: MIT Press, 427--452.
  153. ↑ 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.
  154. ↑ Lawvere, F. W., 1965, Algebraic Theories, Algebraic Categories, and Algebraic Functors, in Theory of Models , Amsterdam: North Holland, 413--418.
  155. ↑ Lawvere, F. W., 1966, The Category of Categories as a Foundation for Mathematics, in Proceedings of the Conference on Categorical Algebra, La Jolla , New York: Springer-Verlag, 1--21.
  156. ↑ Lawvere, F. W., 1969a, Diagonal Arguments and Cartesian Closed Categories, in Category Theory, Homology Theory, and their Applications II , Berlin: Springer, 134--145.
  157. ↑ Lawvere, F. W., 1969b, Adjointness in Foundations, Dialectica , 23, 281--295.
  158. ↑ Lawvere, F. W., 1970, Equality in Hyper-doctrines and Comprehension Schema as an Adjoint Functor, Applications of Categorical Algebra , Providence: AMS, 1-14.
  159. ↑ Lawvere, F. W., 1971, Quantifiers and Sheaves, Actes du Congr\'Es International des Math\'Ematiciens, Tome 1, Paris: Gauthier-Villars, 329--334.
  160. ↑ Lawvere, F. W., 1972, "Introduction", in Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics , 274, Springer-Verlag, 1--12.
  161. ↑ Lawvere, F. W., 1975, Continuously Variable Sets: Algebraic Geometry = Geometric Logic, in Proceedings of the Logic Colloquium Bristol 1973, Amsterdam: North Holland, 135--153.
  162. ↑ Lawvere, F. W., 1976, Variable Quantities and Variable Structures in Topoi, in Algebra, Topology, and Category Theory, New York: Academic Press , 101--131.
  163. ↑ Lawvere, F. W.: 1963, Functorial Semantics of Algebraic Theories, Proc. Natl. Acad. Sci. USA, Mathematics , 50 : 869-872.
  164. ↑ Lawvere, F. W.: 1969, Closed Cartesian Categories ., Lecture held as a guest of the Romanian Academy of Sciences, Bucharest.
  165. ↑ Lawvere, F. W., 1992. Categories of Space and of Quantity, The Space of Mathematics, Foundations of Communication and Cognition.
  166. ↑ Lawvere, F. W., 2002, Categorical Algebra for Continuum Micro-Physics, Journal of Pure and Applied Algebra, 175, no. 1--3, 267--287.
  167. ↑ Lawvere, F. W., 2003, Foundations and Applications: Axiomatization and Education. New Programs and Open Problems in the Foundation of Mathematics, Bulletin of Symbolic Logic , 9, 2, 213--224.
  168. ↑ Leinster, T., 2002, A Survey of Definitions of n-categories, in Theory and Applications of Categories , (electronic), 10 , 1--70.
  169. ↑ Li, M. and P. Vitanyi: 1997, An introduction to Kolmogorov Complexity and its Applications , Springer Verlag: New York.
  170. ↑ Lubkin, S., 1960. Imbedding of abelian categories., Trans. Amer. Math. Soc. , 97 : 410-417.
  171. ↑ K. C. H. Mackenzie : Lie Groupoids and Lie Algebroids in Differential Geometry, LMS Lect. Notes 124 , Cambridge University Press, 1987
  172. ↑ MacLane, S.: 1948. Groups, categories, and duality., Proc. Natl. Acad. Sci.U.S.A , 34 : 263-267.
  173. ↑ MacLane, S., 1969, Foundations for Categories and Sets, in Category Theory, Homology Theory and their Applications II , Berlin: Springer, 146--164.
  174. ↑ MacLane, S., 1971, Categorical algebra and Set-Theoretic Foundations, in Axiomatic Set Theory , Providence: AMS, 231--240.
  175. ↑ Mac Lane, S., 1975, Sets, Topoi, and Internal Logic in Categories, in Studies in Logic and the Foundations of Mathematics , 80, Amsterdam: North Holland, 119-134.
  176. ↑ Mac Lane, S., 1981, Mathematical Models: a Sketch for the Philosophy of Mathematics, American Mathematical Monthly , 88, 7, 462--472.
  177. ↑ Mac Lane, S., 1986, Mathematics, Form and Function , New York: Springer.
  178. ↑ Mac Lane, S., 1988, Concepts and Categories in Perspective, in A Century of Mathematics in America , Part I, Providence: AMS, 323--365.
  179. ↑ Mac Lane, S., 1989, The Development of Mathematical Ideas by Collision: the Case of Categories and Topos Theory, in Categorical Topology and its Relation to Analysis, Algebra and Combinatorics , Teaneck: World Scientific, 1--9.
  180. ↑ MacLane, S., 1950, Dualities for Groups, Bulletin of the American Mathematical Society , 56, 485--516.
  181. ↑ Mac Lane, S., 1996, Structure in Mathematics. Mathematical Structuralism., Philosophia Mathematica, 4, 2, 174-183.
  182. ↑ Mac Lane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.
  183. ↑ Mac Lane, S., 1997, Categorical Foundations of the Protean Character of Mathematics., Philosophy of Mathematics Today, Dordrecht: Kluwer, 117-122.
  184. ↑ Mac Lane, S., and I.~Moerdijk. Sheaves and Geometry in Logic: A First Introduction to Topos Theory , Springer-Verlag, 1992.
  185. ↑ Majid, S.: 1995, Foundations of Quantum Group Theory , Cambridge Univ. Press: Cambridge, UK.
  186. ↑ Majid, S.: 2002, A Quantum Groups Primer , Cambridge Univ.Press: Cambridge, UK.
  187. ↑ 187.0 187.1 Makkai, M. and Par\'e, R., 1989, Accessible Categories: the Foundations of Categorical Model Theory, Contemporary Mathematics 104, Providence: AMS. Cite error: Invalid <ref> tag; name "MM-RG95" defined multiple times with different content
  188. ↑ Makkai, M. and Reyes, G., 1977, First-Order Categorical Logic , Springer Lecture Notes in Mathematics 611, New York: Springer.
  189. ↑ Makkai, M., 1998, Towards a Categorical Foundation of Mathematics, in Lecture Notes in Logic , 11, Berlin: Springer, 153--190.
  190. ↑ Makkai, M., 1999, On Structuralism in Mathematics, in Language, Logic and Concepts , Cambridge: MIT Press, 43--66.
  191. ↑ Mallios, A. and I. Raptis: 2003, Finitary, Causal and Quantal Vacuum Einstein Gravity, Int. J. Theor. Phys. 42 : 1479.
  192. ↑ Manders, K.L.: 1982, On the space-time ontology of physical theories, Philosophy of Science 49 no. 4: 575--590.
  193. ↑ Marquis, J.-P., 1993, Russell's Logicism and Categorical Logicisms, in Russell and Analytic Philosophy , A. D. Irvine \& G. A. Wedekind, (eds.), Toronto, University of Toronto Press, 293--324.
  194. ↑ Marquis, J.-P., 1995, Category Theory and the Foundations of Mathematics: Philosophical Excavations., Synthese , 103, 421--447.
  195. ↑ Marquis, J.-P., 2000, Three Kinds of Universals in Mathematics?, in Logical Consequence: Rival Approaches and New Studies in Exact Philosophy: Logic, Mathematics and Science , Vol. II, B. Brown and J. Woods, eds., Oxford: Hermes, 191-212, 2000 ,
  196. ↑ Marquis, J.-P., 2006, Categories, Sets and the Nature of Mathematical Entities, in: The Age of Alternative Logics. Assessing philosophy of logic and mathematics today , J. van Benthem, G. Heinzmann, Ph. Nabonnand, M. Rebuschi, H.Visser, eds., Springer,181-192.
  197. ↑ Martins, J. F and T. Porter: 2004, On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups,  
  198. ↑ May, J.P. 1999, A Concise Course in Algebraic Topology , The University of Chicago Press: Chicago.
  199. ↑ Mc Larty, C., 1991, Axiomatizing a Category of Categories, Journal of Symbolic Logic , 56, no. 4, 1243-1260.
  200. ↑ Mc Larty, C., 1992, Elementary Categories, Elementary Toposes, Oxford: Oxford University Press.
  201. ↑ Mc Larty, C., 1994, Category Theory in Real Time, Philosophia Mathematica , 2, no. 1, 36-44.
  202. ↑ Misra, B., I. Prigogine and M. Courbage.: 1979. Lyapounov variables: Entropy and measurement in quantum mechanics, Proc. Natl. Acad. Sci. USA 78 (10): 4768-4772.
  203. ↑ Mitchell, B.: 1965, Theory of Categories , Academic Press:London.
  204. ↑ Mitchell, B.: 1964, The full imbedding theorem. Amer. J. Math . 86 : 619-637.
  205. ↑ Moerdijk, I. \& Palmgren, E., 2002, Type Theories, Toposes and Constructive Set Theory: Predicative Aspects of AST., Annals of Pure and Applied Logic, 114, no. 1--3, 155-201.
  206. ↑ Moerdijk, I., 1998, Sets, Topoi and Intuitionism., Philosophia Mathematica , 6, no. 2, 169-177.
  207. ↑ I. Moerdijk : Classifying toposes and foliations, {\it Ann. Inst. Fourier, Grenoble} 41 , 1 (1991) 189-209.
  208. ↑ I. Moerdijk : Introduction to the language of stacks and gerbes, arXiv:math.AT/0212266 (2002).
  209. ↑ Morita, K. 1962. Category isomorphism and endomorphism rings of modules, Trans. Amer. Math. Soc. , 103 : 451-469.
  210. ↑ Morita, K. , 1970. Localization in categories of modules. I., Math. Z. , 114 : 121-144.
  211. ↑ M. A. Mostow : The differentiable space structure of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Diff. Geom. 14 (1979) 255-293.
  212. ↑ Oberst, U.: 1969, Duality theory for Grothendieck categories., Bull. Amer. Math. Soc. 75 : 1401-1408.
  213. ↑ Oort, F.: 1970. On the definition of an abelian category. Proc. Roy. Neth. Acad. Sci . 70 : 13-02.
  214. ↑ Ore, O., 1931, Linear equations on non-commutative fields, Ann. Math. 32 : 463-477.
  215. ↑ Penrose, R.: 1994, Shadows of the Mind , Oxford University Press: Oxford.
  216. ↑ Plymen, R.J. and P. L. Robinson: 1994, Spinors in Hilbert Space , Cambridge Tracts in Math. 114 , Cambridge Univ. Press, Cambridge.
  217. ↑ Pareigis, B., 1970, Categories and Functors, New York: Academic Press.
  218. ↑ Pedicchio, M. C. \& Tholen, W., 2004, Categorical Foundations, Cambridge: Cambridge University Press.
  219. ↑ Pitts, A. M., 2000, Categorical Logic, in Handbook of Logic in Computer Science , Vol.5, Oxford: Oxford University Press, 39--128.
  220. ↑ Plotkin, B., 2000, Algebra, Categories and Databases, in Handbook of Algebra , Vol. 2, Amsterdam: Elsevier, 79--148.
  221. ↑ Popescu, N.: 1973, Abelian Categories with Applications to Rings and Modules. New York and London: Academic Press., 2nd edn. 1975. (English translation by I.C. Baianu) .
  222. ↑ Pradines, J.: 1966, Th\'eorie de Lie pour les groupoides diff\'erentiable, relation entre propri\'etes locales et globales, C. R. Acad Sci. Paris S\'er. A 268 : 907-910.
  223. ↑ Pribram, K. H.: 2000, Proposal for a quantum physical basis for selective learning, in (Farre, ed.) Proceedings ECHO IV 1-4.
  224. ↑ Prigogine, I.: 1980, From Being to Becoming : Time and Complexity in the Physical Sciences , W. H. Freeman and Co.: San Francisco.
  225. ↑ Raptis, I. and R. R. Zapatrin: 2000, Quantisation of discretized spacetimes and the correspondence principle, Int. Jour. Theor. Phys. 39 : 1.
  226. ↑ Raptis, I.: 2003, Algebraic quantisation of causal sets, Int. Jour. Theor. Phys. 39 : 1233.
  227. ↑ I. Raptis : Quantum space--time as a quantum causal set, arXiv:gr--qc/0201004.
  228. ↑ Reyes, G. and Zolfaghari, H., 1991, Topos-theoretic Approaches to Modality, Category Theory (Como 1990), Lecture Notes in Mathematics , 1488, Berlin: Springer, 359--378.
  229. ↑ Reyes, G. andZolfaghari, H., 1996, Bi-Heyting Algebras, Toposes and Modalities, Journal of Philosophical Logic , 25, no. 1, 25--43.
  230. ↑ Reyes, G., 1974, From Sheaves to Logic, in Studies in Algebraic Logic , A. Daigneault, ed., Providence: AMS.
  231. ↑ Reyes, G., 1991, A Topos-theoretic Approach to Reference and Modality., Notre Dame Journal of Formal Logic , 32, no. 3, 359-391.
  232. ↑ M. A. Rieffel : Group C*--algebras as compact quantum metric spaces, Documenta Math. 7 (2002), 605-651.
  233. ↑ Roberts, J. E.: 2004, More lectures on algebraic quantum field theory, in A. Connes, et al. Noncommutative Geometry , Springer: Berlin and New York.
  234. ↑ Rodabaugh, S. E. \& Klement, E. P., eds., Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic, 20, Dordrecht: Kluwer.
  235. ↑ Rota, G. C. : On the foundation of combinatorial theory, I. The theory of M\"obius functions, \emph{Zetschrif f\"ur Wahrscheinlichkeitstheorie} 2 (1968), 340.
  236. ↑ Rovelli, C.: 1998, Loop Quantum Gravity, in N. Dadhich, et al. Living Reviews in Relativity (refereed electronic journal)
  237. ↑ Schr\"odinger E.: 1945, What is Life? , Cambridge University Press: Cambridge, UK.
  238. ↑ Scott, P. J., 2000, Some Aspects of Categories in Computer Science , Handbook of Algebra, Vol. 2, Amsterdam: North Holland, 3--77.
  239. ↑ Seely, R. A. G., 1984, Locally Cartesian Closed Categories and Type Theory, Mathematical Proceedings of the Cambridge Mathematical Society , 95, no. 1, 33-48.
  240. ↑ Shapiro, S., 2005, Categories, Structures and the Frege-Hilbert Controversy: the Status of Metamathematics, Philosophia Mathematica , 13, 1, 61--77.
  241. ↑ Sorkin, R.D.: 1991, Finitary substitute for continuous topology, Int. J. Theor. Phys. 30 No. 7.: 923--947.
  242. ↑ Smolin, L.: 2001, Three Roads to Quantum Gravity , Basic Books: New York.
  243. ↑ Spanier, E. H.: 1966, Algebraic Topology , McGraw Hill: New York.
  244. ↑ Stapp, H.: 1993, Mind, Matter and Quantum Mechanics , Springer Verlag: Berlin--Heidelberg--New York.
  245. ↑ Stewart, I. and Golubitsky, M. : 1993. Fearful Symmetry: Is God a Geometer? , Blackwell: Oxford, UK.
  246. ↑ Szabo, R. J.: 2003, Quantum field theory on non-commutative spaces, Phys. Rep. 378 : 207--209.
  247. ↑ Taylor, P., 1996, Intuitionistic sets and Ordinals, Journal of Symbolic Logic , 61 : 705-744.
  248. ↑ Taylor, P., 1999, Practical Foundations of Mathematics , Cambridge: Cambridge University Press.
  249. ↑ Unruh, W.G.: 2001, Black holes, dumb holes, and entropy, in C. Callender and N. Hugget (eds. ) Physics Meets Philosophy at the Planck scale , Cambridge University Press, pp. 152-173.
  250. ↑ Van der Hoeven, G. and Moerdijk, I., 1984a, Sheaf Models for Choice Sequences, Annals of Pure and Applied Logic, 27, no. 1, 63--107.
  251. ↑ V\'arilly, J. C.: 1997, An introduction to noncommutative geometry \\ arXiv:physics/9709045 London.
  252. ↑ von Neumann, J.: 1932, Mathematische Grundlagen der Quantenmechanik , Springer: Berlin.
  253. ↑ Weinstein, A.: 1996, Groupoids : unifying internal and external symmetry, Notices of the Amer. Math. Soc. 43 : 744--752.
  254. ↑ Wess J. and J. Bagger: 1983, Supersymmetry and Supergravity , Princeton University Press: Princeton, NJ.
  255. ↑ Weinberg, S.: 1995, The Quantum Theory of Fields vols. 1 to 3, Cambridge Univ. Press.
  256. ↑ Wheeler, J. and W. Zurek: 1983, Quantum Theory and Measurement , Princeton University Press: Princeton, NJ.
  257. ↑ Whitehead, J. H. C.: 1941, On adding relations to homotopy groups, Annals of Math. 42 (2): 409--428.
  258. ↑ Woit, P.: 2006, Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Laws , Jonathan Cape.
  259. ↑ Wood, R.J., 2004, Ordered Sets via Adjunctions, In: Categorical Foundations , M. C. Pedicchio \& W. Tholen, eds., Cambridge: Cambridge University Press.
Retrieved from "https://en.wikiversity.org/w/index.php?title=PlanetPhysics/Axiomatics_and_Categorical_Foundations_of_Mathematical_Physics&oldid=2562957"