A Bibliography for Axiomatic Theories and Categorical Foundations of Mathematical Physics and Mathematics

a. Foundations of Mathematics, Logics and Formal Logics: Axiomatics, Categories, Topoi and Higher Dimensional Algebra

Bibliography

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2

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11

Bell, J. L. 1981. Category Theory and the Foundations of Mathematics, British Journal for the Philosophy of Science, 32, 349–358.

12

Bell, J. L., 1982. Categories, Toposes and Sets, Synthese, 51(3): 293–337.

13

Blass, A. and Scedrov, A., 1983, Classifying Topoi and Finite Forcing , Journal of Pure and Applied Algebra, 28, 111–140.

14

Blass, A. and Scedrov, A., 1989, Freyd's Model for the Independence of the Axiom of Choice, Providence: AMS.

15

Blass, A. and Scedrov, A., 1992. Complete Topoi Representing Models of Set Theory, Annals of Pure and Applied Logic , 57, no. 1, 1–26.

16

Blass, A., 1984, The Interaction Between Category Theory and Set Theory., Mathematical Applications of Category Theory, 30, Providence: AMS, 5–29.

17

Blute, R. and Scott, P., 2004, Category Theory for Linear Logicians., in Linear Logic in Computer Science

18

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.

19

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.

20

Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. Géom. Diff. 17, 343-362.

21

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.

22

Ronald Brown et al.: Non-Abelian Algebraic Topology, vols. I and II. 2010, 620 pages with Index. (March 4, 2010-in press: Springer): Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical higher homotopy groupoids

23

Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80: 1-34.

24

Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179, 291-317.

25

Bunge, M., 1984, Toposes in Logic and Logic in Toposes, Topoi, 3, no. 1, 13-22.

26

Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math, 179: 291-317.

27

Ehresmann, C.: 1965, Catégories et Structures, Dunod, Paris.

27

Ehresmann, C.: 1966, Trends Toward Unity in Mathematics., Cahiers de Topologie et Geometrie Differentielle 8: 1-7.

b. Universal Algebra, Classes of Algebraic Structures and Homology; Abelian and Non-Abelian theories; Algebraic geometry and Noncommutative Geometry

Bibliography

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R. Brown. 2008. Higher Dimensional Algebra Preprint as pdf and ps docs. at arXiv:math/0212274v6 [math.AT]

3

Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr., 71: 273-286.

4

Cartan, H. and Eilenberg, S. 1956. Homological Algebra, Princeton Univ. Press: Pinceton.

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Chevalley, C. 1946. The theory of Lie groups. Princeton University Press, Princeton NJ.

6

M. Chaician and A. Demichev. 1996. Introduction to Quantum Groups, World Scientific .

7

Cohen, P.M. 1965. Universal Algebra, Harper and Row: New York, London and Tokyo.

8

Connes A 1994. Noncommutative geometry. Academic Press: New York.

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Croisot, R. and Lesieur, L. 1963. Algèbre noethérienne non-commutative., Gauthier-Villard: Paris.

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Crole, R.L., 1994, Categories for Types, Cambridge: Cambridge University Press.

11

Dieudonné, J. and Grothendieck, A., 1960, [1971], Éléments de Géométrie Algébrique, Berlin: Springer-Verlag.

12

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.]

13

M. Durdevich : Geometry of quantum principal bundles I, Commun. Math. Phys. 175 (3) (1996), 457–521.

14

M. Durdevich : Geometry of quantum principal bundles II, Rev.Math. Phys. 9 (5) (1997), 531-607.

15

Ehresmann, C.: 1952, Structures locales et structures infinitésimales, C.R.A.S. Paris 274: 587-589.

16

Ehresmann, C.: 1959, Catégories topologiques et catégories différentiables, Coll. Géom. Diff. Glob. Bruxelles, pp.137-150.

17

Ehresmann, C.:1963, Catégories doubles des quintettes: applications covariantes , C.R.A.S. Paris, 256: 1891–1894.

18

Ehresmann, C.: 1984, Oeuvres complètes et commentées: Amiens, 1980-84, edited and commented by Andrée Ehresmann.

19

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.

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Eilenberg, S. & MacLane, S., 1942, Group Extensions and Homology, Annals of Mathematics, 43, 757–831.

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Eilenberg, S. & Steenrod, N., 1952, Foundations of Algebraic Topology, Princeton: Princeton University Press.

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Eilenberg, S.: 1960. Abstract description of some basic functors., J. Indian Math.Soc., 24 :221-234.

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S.Eilenberg. Relations between Homology and Homotopy Groups. Proc.Natl.Acad.Sci.USA (1966),v:10–14.

26

Ellerman, D., 1988, Category Theory and Concrete Universals, Synthese, 28, 409–429.

27

Ezawa,Z.F., G. Tsitsishvilli and K. Hasebe : Noncommutative geometry, extended algebra and Grassmannian solitons in multicomponent Hall systems, (at arXiv:hep–th/0209198).

28

Freyd, P., 2002, Cartesian Logic, Theoretical Computer Science, 278, no. 1–2, 3–21.

29

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.

30

Gablot, R. 1971. Sur deux classes de catégories de Grothendieck. Thesis, Univ. de Lille.

31

Gabriel, P.: 1962, Des catégories abéliennes, Bull. Soc. Math. France 90: 323-448.

32

Gabriel, P. and M.Zisman:. 1967: Category of fractions and homotopy theory, Ergebnesse der math. Springer: Berlin.

33

Gabriel, P. and N. Popescu: 1964, Caractérisation des catégories abéliennes avec générateurs et limites inductives. , CRAS Paris 258: 4188-4191.

34

Galli, A. & Reyes, G. & Sagastume, M., 2000, Completeness Theorems via the Double Dual Functor, Studia Logica, 64, no. 1: 61–81.

35

Gelfan'd, I. and Naimark, M., 1943, On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space, Recueil Mathématique [Matematicheskii Sbornik] Nouvelle Série, 12 [54]: 197–213. [Reprinted in C*–algebras: 1943–1993, in the series Contemporary Mathematics, 167, Providence, R.I.: American Mathematical Society, 1994.]

36

Ghilardi, S. & Zawadowski, M., 2002, Sheaves, Games & Model Completions: A Categorical Approach to Nonclassical Propositional Logics, Dordrecht: Kluwer.

37

Ghilardi, S., 1989, Presheaf Semantics and Independence Results for some Non-classical first-order logics, Archive for Mathematical Logic, 29, no. 2, 125–136.

38

Goblot, R., 1968, Catégories modulaires , C. R. Acad. Sci. Paris, Série A., 267: 381–383.

39

Goblot, R., 1971, Sur deux classes de catégories de Grothendieck, Thèse., Univ. Lille, 1971.

40

Goldblatt, R., 1979, Topoi: The Categorical Analysis of Logic, Studies in logic and the foundations of mathematics, Amsterdam: Elsevier North-Holland Publ. Comp.

41

Goldie, A. W., 1964, Localization in non-commutative noetherian rings, J.Algebra, 1: 286-297.

42

Godement,R. 1958. Théorie des faisceaux. Hermann: Paris.

43

Gray, C. W.: 1965. Sheaves with values in a category.,Topology, 3: 1-18.

44

Grothendieck, A.: 1971, Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées et descente, Lecture Notes in Math. 224, Springer–Verlag: Berlin.

45

Grothendieck, A.: 1957, Sur quelque point d-algèbre homologique. , Tohoku Math. J., 9: 119-121.

46

Grothendieck, A. and J. Dieudoné.: 1960, Eléments de geometrie algébrique., Publ. Inst. des Hautes Etudes de Science, 4.

47

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

48

Hardie, K.A. K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Applied Cat. Structures 8 (2000), 209-234.

49

Hatcher, W. S., 1982, The Logical Foundations of Mathematics, Oxford: Pergamon Press.

50

Heller, A. :1958, Homological algebra in Abelian categories., Ann. of Math. 68: 484-525.

51

Heller, A. and K. A. Rowe.:1962, On the category of sheaves., Amer J. Math. 84: 205-216.

52

Hellman, G., 2003, "Does Category Theory Provide a Framework for Mathematical Structuralism?", Philosophia Mathematica, 11, 2, 129–157.

53

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.

54

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.

55

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.

56

Higgins, P. J.: 2005, Categories and groupoids, Van Nostrand Mathematical Studies: 32, (1971); Reprints in Theory and Applications of Categories, No. 7: 1-195.

57

Higgins, Philip J. Thin elements and commutative shells in cubical -categories. Theory Appl. Categ. 14 (2005), No. 4, 60–74 (electronic). msc: 18D05.

58

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.

59

Hyland, J.M.E., 1982, The Effective Topos, Studies in Logic and the Foundations of Mathematics, 110, Amsterdam: North Holland, 165–216.

60

Hyland, J. M..E., 1988, A Small Complete Category, Annals of Pure and Applied Logic, 40, no. 2, 135–165.

61

Hyland, J. M .E., 1991, First Steps in Synthetic Domain Theory, Category Theory (Como 1990), Lecture Notes in Mathematics, 1488, Berlin: Springer, 131-156.

62

Hyland, J. M.E., 2002, Proof Theory in the Abstract, Annals of Pure and Applied Logic, 114, no. 1–3, 43–78.

63

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64

Jacobs, B., 1999, Categorical Logic and Type Theory, Amsterdam: North Holland.

65

Johnstone, P. T., 1977, Topos Theory, New York: Academic Press.

66

Johnstone, P. T., 1979a, Conditions Related to De Morgan's Law, Applications of Sheaves, Lecture Notes in Mathematics, 753, Berlin: Springer, 479–491.

67

Johnstone, P. T., 1981, Tychonoff's Theorem without the Axiom of Choice, Fundamenta Mathematicae, 113, no. 1, 21–35.

68

Johnstone, P. T., 1982, Stone Spaces, Cambridge:Cambridge University Press.

69

Johnstone, P. T., 1985, How General is a Generalized Space?, Aspects of Topology, Cambridge: Cambridge University Press, 77–111.

70

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71

Van Kampen, E. H.: 1933, On the Connection Between the Fundamental Groups of some Related Spaces, Amer. J. Math. 55: 261-267

72

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73

Kleisli, H.: 1962, Homotopy theory in Abelian categories.,Can. J. Math., 14: 139-169.

74

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75

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76

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77

H. Krips : Measurement in Quantum Theory, The Stanford Encyclopedia of Philosophy (Winter 1999 Edition), Edward N. Zalta (ed.),

78

Lam, T. Y., 1966, The category of noetherian modules, Proc. Natl. Acad. Sci. USA, 55: 1038-104.

79

Lambek, J. & Scott, P.J., 1986, Introduction to Higher Order Categorical Logic, Cambridge: Cambridge University Press.

80

Lambek, J., 1968, Deductive Systems and Categories I. Syntactic Calculus and Residuated Categories, Mathematical Systems Theory, 2, 287–318.

81

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.

82

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.

83

Lambek, J., 1989A, On Some Connections Between Logic and Category Theory, Studia Logica, 48, 3, 269–278.

84

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.

85

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.

86

Lambek, J., 2004, What is the world of Mathematics? Provinces of Logic Determined, Annals of Pure and Applied Logic, 126(1-3), 149–158.

87

Lambek, J. and P. J. Scott. Introduction to higher order categorical logic. Cambridge University Press, 1986.

88

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89

Landry, E. & Marquis, J.-P., 2005, Categories in Context: Historical, Foundational and philosophical, Philosophia Mathematica, 13, 1–43.

90

Landry, E., 1999, Category Theory: the Language of Mathematics, Philosophy of Science, 66, 3: supplement, S14–S27.

91

Landsman, N. P.: 1998, Mathematical Topics between Classical and Quantum Mechanics, Springer Verlag: New York.

92

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94

La Palme Reyes, M., et. al., 1999, Count Nouns, Mass Nouns, and their Transformations: a Unified Category-theoretic Semantics, Language, Logic and Concepts, Cambridge: MIT Press, 427–452.

95

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.

96

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

97

Lawvere, F. W.: 1966, The Category of Categories as a Foundation for Mathematics., in Proc. Conf. Categorical Algebra- La Jolla., Eilenberg, S. et al., eds. Springer–Verlag: Berlin, Heidelberg and New York., pp. 1-20.

98

Lawvere, F. W., 1969a, Diagonal Arguments and Cartesian Closed Categories, in Category Theory, Homology Theory, and their Applications II, Berlin: Springer, 134–145.

99

Lawvere, F. W., 1969b, Adjointness in Foundations, Dialectica, 23: 281–295.

100

Lawvere, F. W., 1970, Equality in Hyper doctrines and Comprehension Schema as an Adjoint Functor, Applications of Categorical Algebra, Providence: AMS, 1-14.

101

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102

Lawvere, F. W., 1972, Introduction, in Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, 274, Springer-Verlag, 1-12.

103

Lawvere, F. W., 1975, Continuously Variable Sets: Algebraic Geometry = Geometric Logic, Proceedings of the Logic Colloquium, Bristol 1973, Amsterdam: North Holland, 135-153.

104

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105

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106

Lawvere, F. W., 1992, Categories of Space and of Quantity, The Space of Mathematics, Foundations of Communication and Cognition, Berlin: De Gruyter, 14-30.

107

Lawvere, F. W., 1994b, Tools for the Advancement of Objective Logic: Closed Categories and Toposes, The Logical Foundations of Cognition, Vancouver Studies in Cognitive Science, 4, Oxford: Oxford University Press, 43–56.

108

Lawvere, F. W., 2000, Comments on the Development of Topos Theory, Development of Mathematics 1950-2000, Basel: Birkh'́auser, 715–734.

109

Lawvere, F. W., 2002, Categorical Algebra for Continuum Micro-Physics, Journal of Pure and Applied Algebra, 175, no. 1–3, 267–287.

110

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.

111

Leinster, T., 2002, A Survey of Definitions of n-categories, in Theory and Applications of Categories, (electronic), 10, 1–70.

112

Li, M. and P. Vitanyi: 1997, An introduction to Kolmogorov Complexity and its Applications, Springer Verlag: New York.

113

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114

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115

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116

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117

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118

MacLane, S., 1971, Categorical algebra and Set-Theoretic Foundations, in Axiomatic Set Theory, Providence: AMS, 231–240.

119

MacLane, S., 1975, Sets, Topoi, and Internal Logic in Categories, Studies in Logic and the Foundations of Mathematics, 80, Amsterdam: North Holland, 119–134.

120

MacLane, S., 1986, Mathematics, Form and Function, New York: Springer.

121

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122

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123

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124

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125

MacLane, S., 1996, Structure in Mathematics. Mathematical Structuralism., Philosophia Mathematica, 4, 2, 174-183.

126

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127

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128

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129

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130

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131

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129

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132

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133

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134

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135

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139

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144

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145

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146

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155

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156

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158

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159

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160

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162

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164

Reyes, G. & Zolfaghari, H., 1991, Topos-theoretic Approaches to Modality,in Category Theory (Como 1990), Lecture Notes in Mathematics, 1488, Berlin: Springer, 359–378.

165

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