A list of references in: Algebraic topology, quantum algebraic topology, n-logic algebraic categories, Theory of Categories, functors, natural transformations, Topoi and categorical ontology.

This is an extensive, but not intended to be comprehensive, list of selected references for several areas of both abstract and applied mathematics relevant to quantum algebraic topology. A more extensive bibliography on category theory can be found on the web at the Plato, Stanford Encyclopedia of Philosophy web site.

Bibliography

1

Adámek, J.. et al., Locally Presentable and Accessible Categories, Cambridge: Cambridge University Press (1994).

2

Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, Boston–Basel–Berlin (2003).

3

Atiyah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France, 84: 307–317.

3

Auslander, M. 1965. Coherent Functors. Proc. Conf. Cat. Algebra, La Jolla, 189–231.

4

Awodey, S. & Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168–1182.

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.

5

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.

6

Awodey, S., 1996, Structure in Mathematics and Logic: A Categorical Perspective, Philosophia Mathematica, 3, 209-237.

7

Awodey, S., 2004, An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism., Philosophia Mathematica, 12, 54-64.

8

Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.

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.

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

21

Baianu, I.C., H. S. Gutowsky, and E. Oldfield: 1984, Proc. Natl. Acad. Sci. USA, 81(12): 3713-3717.

20

Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic Networks, FASEB Proceedings 43, 917.

22

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.

23

Baianu, I. C.: 1987b, Molecular Models of Genetic and Organismic Structures, in Proceed. Relational Biology Symp. Argentina; CERN Preprint No.EXT-2004-067 .

24

Baianu, I.C.: 2004a. Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint: w. Cogprints at Sussex Univ.

25

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).

26

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.

27

Baianu, I.C.: 2004a, Quantum Nano–Automata (QNA): Microphysical Measurements with Microphysical QNA Instruments, CERN Preprint EXT–2004–125.

28

Baianu, I. C.: 2004b, Quantum Interactomics and Cancer Mechanisms, Preprint 00001978 .

29

Baianu, I. C.: 2006, Robert Rosen's Work and Complex Systems Biology, Axiomathes 16(1–2):25–34.

30

Baianu, I. C., Brown, R. and J. F. Glazebrook: 2006, Quantum Algebraic Topology and Field Theories. Preprint

31

Baianu, I.C.: 2008, Translational Genomics and Human Cancer Interactomics, (invited Review, submitted in November 2007 to Translational Oncogenomics).

32

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.

33

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.

34

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

35

M. Barr and C. Wells. Toposes, Triples and Theories. Montreal: McGill University, 2000.

36

Barr, M. & Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.

37

Barr, M. & Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.

38

Batanin, M., 1998, Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-Categories", Advances in Mathematics, 136, 39–103.

39

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

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.

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

73

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

74

Lawvere, F. W., 1971, “Quantifiers and Sheaves.”, Actes du Congrés International des Mathématiciens, Tome 1, Paris: Gauthier-Villars, 329–334.

75

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

76

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

77

Lawvere, F. W., 1976, “Variable Quantities and Variable Structures in Topoi.”, Algebra, Topology, and Category Theory, New York: Academic Press, 101–131.

78

Lawvere, F. W. & Schanuel, S., 1997, Conceptual Mathematics: A First Introduction to Categories, Cambridge: Cambridge University Press.

79

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.

80

Lawvere, F. W.: 1963, Functorial Semantics of Algebraic Theories, Proc. Natl. Acad. Sci. USA, Mathematics, 50: 869-872.

81

Lawvere, F. W.: 1969, Closed Cartesian Categories., Lecture held as a guest of the Romanian Academy of Sciences, Bucharest.

82

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

83

Lawvere, F. W., 1994a, “Cohesive Toposes and Cantor's lauter Ensein.”, Philosophia Mathematica, 2, 1, 5–15.

84

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.

85

Lawvere, H. W (ed.), 1995. Springer Lecture Notes in Mathematics 274,:13–42.

86

Lawvere, F. W., 2000, “Comments on the Development of Topos Theory.”, Development of Mathematics 1950-2000, Basel: Birkhäuser, 715–734.

87

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

88

Lawvere, F. W. & Rosebrugh, R., 2003, Sets for Mathematics, Cambridge: Cambridge University Press.

89

Lawvere, F. W., 2003, “Foundations and Applications: Axiomatization and Education. New Programs and Open Problems in the Foundation of Mathematics.”, Bullentin of Symbolic Logic, 9, 2, 213–224.

80

Lawvere, F.W., 1963, “Functorial Semantics of Algebraic Theories.”, Proceedings of the National Academy of Sciences U.S.A., 50, 869–872.

90

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

91

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

92

Löfgren, L.: 1968, An Axiomatic Explanation of Complete Self-Reproduction, Bulletin of Mathematical Biophysics, 30: 317-348

93

Lubkin, S., 1960. Imbedding of abelian categories., Trans. Amer. Math. Soc., 97: 410-417.

94

Luisi, P. L. and F. J. Varela: 1988, Self-replicating micelles a chemical version of a minimal autopoietic system. Origins of Life and Evolution of Biospheres 19(6):633–643.

95

K. C. H. Mackenzie : Lie Groupoids and Lie Algebroids in Differential Geometry, LMS Lect. Notes 124, Cambridge University Press, 1987

96

MacLane, S.: 1948. Groups, categories, and duality., Proc. Natl. Acad. Sci.U.S.A, 34: 263-267.

97

MacLane, S., 1969, “Foundations for Categories and Sets.', Category Theory, Homology Theory and their Applications II, Berlin: Springer, 146–164.

98

MacLane, S., 1969, “One Universe as a Foundation for Category Theory.”, Reports of the Midwest Category Seminar III, Berlin: Springer, 192–200.

99

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

100

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

101

MacLane, S., 1981, “Mathematical Models: a Sketch for the Philosophy of Mathematics.”, American Mathematical Monthly, 88, 7, 462–472.

102

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

103

MacLane, S., 1988, “Concepts and Categories in Perspective”, A Century of Mathematics in America, Part I, Providence: AMS, 323–365.

104

MacLane, S., 1989, “The Development of Mathematical Ideas by Collision: the Case of Categories and Topos Theory.”, Categorical Topology and its Relation to Analysis, Algebra and Combinatorics, Teaneck: World Scientific, 1–9.

105

S. Maclane and I. Moerdijk : Sheaves in Geometry and Logic- A first Introduction to Topos Theory, Springer Verlag, New York, 1992.

106

MacLane, S., 1950, “Dualities for Groups”, Bulletin of the American Mathematical Society, 56, 485-516.

107

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

108

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

109

MacLane, S., 1997, Categorical Foundations of the Protean Character of Mathematics., Philosophy of Mathematics Today, Dordrecht: Kluwer, 117–122.

110

MacLane, S., and I. Moerdijk. Sheaves and Geometry in Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.

111

Majid, S.: 1995, Foundations of Quantum Group Theory, Cambridge Univ. Press: Cambridge, UK.

112

Majid, S.: 2002, A Quantum Groups Primer, Cambridge Univ.Press: Cambridge, UK.

113

Makkai, M. & Paré, R., 1989, Accessible Categories: the Foundations of Categorical Model Theory, Contemporary Mathematics 104, Providence: AMS.

114

Makkai, M., 1998, Towards a Categorical Foundation of Mathematics, Lecture Notes in Logic, 11, Berlin: Springer, 153–190.

115

Makkai, M., 1999, “On Structuralism in Mathematics”, in Language, Logic and Concepts, Cambridge: MIT Press, 43–66.

116

Makkai, M. & Reyes, G., 1977, First-Order Categorical Logic, Springer Lecture Notes in Mathematics 611, New York: Springer.

114

Makkai, M., 1998, “Towards a Categorical Foundation of Mathematics.”, Lecture Notes in Logic, 11, Berlin: Springer, 153–190.

115

Makkai, M., 1999, “On Structuralism in Mathematics, Language, Logic and Concepts.”, Cambridge: MIT Press, 43–66.

113

Makkei, M. & Reyes, G., 1995, “Completeness Results for Intuitionistic and Modal Logic in a Categorical Setting.”, Annals of Pure and Applied Logic, 72, 1, 25–101.

117

Mallios, A. and I. Raptis: 2003, Finitary, Causal and Quantal Vacuum Einstein Gravity, Int. J. Theor. Phys. 42: 1479.

118

Manders, K.L.: 1982, On the space-time ontology of physical theories, Philosophy of Science 49 no. 4: 575–590.

119

Marquis, J.-P., 1993, “Russell's Logicism and Categorical Logicisms”, Russell and Analytic Philosophy, A. D. Irvine & G. A. Wedekind, (eds.), Toronto, University of Toronto Press, 293–324.

120

Marquis, J.-P., 1995, Category Theory and the Foundations of Mathematics: Philosophical Excavations., Synthese, 103, 421–447.

121

Marquis, J.-P., 2000, “Three Kinds of Universals in Mathematics?”, Logical Consequence: Rival Approaches and New Studies in Exact Philosophy: Logic, Mathematics and Science, Vol. 2, Oxford: Hermes, 191–212.

122

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.

123

Martins, J. F and T. Porter: 2004, On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups, math.QA/0608484

124

Maturana, H. R. and F. J. Varela: 1980, Autopoiesis and Cognition-The Realization of the Living, Boston Studies in the Philosophy of Science Vol. 42, Reidel Pub. Co.: Dordrecht.

125

May, J.P. 1999, A Concise Course in Algebraic Topology, The University of Chicago Press: Chicago.

126

McCulloch, W. and W. Pitt.: 1943, A logical Calculus of Ideas Immanent in Nervous Activity., Bull. Math. Biophysics, 5: 115-133.

127

Mc Larty, C., 1986, Left Exact Logic, Journal of Pure and Applied Algebra, 41, no. 1, 63-66.

128

Mc Larty, C., 1991, “Axiomatizing a Category of Categories.”, Journal of Symbolic Logic, 56, no. 4, 1243-1260.

129

Mc Larty, C., 1992, “Elementary Categories, Elementary Toposes”, Oxford: Oxford University Press.

130

Mc Larty, C., 1994, “Category Theory in Real Time.”, Philosophia Mathematica, 2, no. 1, 36-44.

131

Mc Larty, C., 2004, “Exploring Categorical Structuralism”, Philosophia Mathematica, 12, 37-53.

132

Mc Larty, C., 2005, “Learning from Questions on Categorical Foundations”, Philosophia Mathematica, 13, 1, 44–60.

133

Misra, B., I. Prigogine and M. Courbage.: 1979, Lyaponouv variables: Entropy and measurement in quantum mechanics, Proc. Natl. Acad. Sci. USA 78 (10): 4768–4772.

134

Mitchell, B.: 1965, Theory of Categories, Academic Press:London.

135

Mitchell, B.: 1964, The full imbedding theorem. Amer. J. Math. 86: 619-637.

136

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.

137

Moerdijk, I., 1998, Sets, Topoi and Intuitionism., Philosophia Mathematica, 6, no. 2, 169ÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÂÂ�177.

138

I. Moerdijk : Classifying toposes and foliations, Ann. Inst. Fourier, Grenoble 41, 1 (1991) 189-209.

139

I. Moerdijk : Introduction to the language of stacks and gerbes, arXiv:math.AT/0212266 (2002).

140

Morita, K. 1962. Category isomorphism and endomorphism rings of modules, Trans. Amer. Math. Soc., 103: 451-469.

141

Morita, K. , 1970. Localization in categories of modules. I., Math. Z., 114: 121-144.

142

M. A. Mostow : The differentiable space structure of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Diff. Geom. 14 (1979) 255-293.

143

Oberst, U.: 1969, Duality theory for Grothendieck categories., Bull. Amer. Math. Soc. 75: 1401-1408.

144

Oort, F.: 1970. On the definition of an abelian category. Proc. Roy. Neth. Acad. Sci. 70: 13-02.

145

Ore, O., 1931, Linear equations on non-commutative fields, Ann. Math. 32: 463-477.

146

Penrose, R.: 1994, Shadows of the Mind, Oxford University Press: Oxford.

147

Plymen, R.J. and P. L. Robinson: 1994, Spinors in Hilbert Space, Cambridge Tracts in Math. 114, Cambridge Univ. Press, Cambridge.

148

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).

149

Pareigis, B., 1970, Categories and Functors, New York: Academic Press.

150

Pedicchio, M. C. & Tholen, W., 2004, Categorical Foundations, Cambridge: Cambridge University Press.

151

Peirce, B., 1991, Basic Category Theory for Computer Scientists, Cambridge: MIT Press.

152

Pitts, A. M., 1989, “Conceptual Completeness for First-order Intuitionistic Logic: an Application of Categorical Logic.”, Annals of Pure and Applied Logic, 41, no. 1, 33–81.

153

Pitts, A. M., 2000, “Categorical Logic”, Handbook of Logic in Computer Science, Vol.5, Oxford: Oxford Unversity Press, 39–128.

154

Plotkin, B., 2000, “Algebra, Categories and Databases”, Handbook of Algebra, Vol. 2, Amsterdam: Elsevier, 79–148.

155

Poli, R.: 2008, Ontology: The Categorical Stance, (in TAO1- Theory and Applications of Ontology: vol.1).

155

Poli, R. (with I.C. Baianu): 2008, Categorical Ontology: the theory of levels, (in TAO1- Theory and Applications of Ontology: vol.1), in press.

156

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).

157

Porter, T.: 2002, Geometric aspects of multiagent sytems, preprint University of Wales-Bangor.

158

Pradines, J.: 1966, Théorie de Lie pour les groupoides différentiable, relation entre propriétes locales et globales, C. R. Acad Sci. Paris Sér. A 268: 907-910.

159

Pribram, K. H.: 1991, Brain and Perception: Holonomy and Structure in Figural processing, Lawrence Erlbaum Assoc.: Hillsdale.

160

Pribram, K. H.: 2000, Proposal for a quantum physical basis for selective learning, in (Farre, ed.) Proceedings ECHO IV 1-4.

161

Prigogine, I.: 1980, From Being to Becoming : Time and Complexity in the Physical Sciences, W. H. Freeman and Co.: San Francisco.

162

Raptis, I. and R. R. Zapatrin: 2000, Quantisation of discretized spacetimes and the correspondence principle, Int. Jour. Theor. Phys. 39: 1.

163

Raptis, I. 2003, Algebraic quantisation of causal sets, Int. Jour. Theor. Phys. 39: 1233.

164

I. Raptis. Quantum space–time as a quantum causal set, arXiv:gr–qc/0201004.

165

Rashevsky, N. 1965, The Representation of Organisms in Terms of Predicates, Bulletin of Mathematical Biophysics 27: 477-491.

166

Rashevsky, N. 1969, Outline of a Unified Approach to Physics, Biology and Sociology., Bulletin of Mathematical Biophysics 31: 159–198.

167

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

168

Reyes, G. & Zolfaghari, H., 1996, “Bi-Heyting Algebras, Toposes and Modalities.”, Journal of Philosophical Logic, 25, no. 1, 25–43.

169

Reyes, G., 1974, “From Sheaves to Logic.”, in Studies in Algebraic Logic, A. Daigneault, ed., Providence: AMS.

170

Reyes, G., 1991, “A Topos-theoretic Approach to Reference and Modality”, Notre Dame Journal of Formal Logic, 32, no. 3, 359-391.

171

M. A. Rieffel : Group C*–algebras as compact quantum metric spaces, Documenta Math. 7 (2002), 605-651.

172

Roberts, J. E.: 2004, More lectures on algebraic quantum field theory, in A. Connes, et al. Noncommutative Geometry, Springer: Berlin and New York.

173

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.

174

Rosen, R.: 1985, Anticipatory Systems, Pergamon Press: New York.

175

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

176

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

177

Rosen, R.: 1987, On Complex Systems, European Journal of Operational Research 30, 129ÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÂÂ�134.

178

G. C. Rota : On the foundation of combinatorial theory, I. The theory of Möbius functions, Zetschrif für Wahrscheinlichkeitstheorie 2 (1968), 340.

179

Rovelli, C.: 1998, Loop Quantum Gravity, in N. Dadhich, et al. Living Reviews in Relativity (refereed electronic journal) ">online download

180

Schrödinger E.: 1967, Mind and Matter in `What is Life?', Cambridge University Press: Cambridge, UK.

181

Schrödinger E.: 1945, What is Life?, Cambridge University Press: Cambridge, UK.

182

Scott, P. J., 2000, Some Aspects of Categories in Computer Science, Handbook of Algebra, Vol. 2, Amsterdam: North Holland, 3–77.

183

Seely, R. A. G., 1984, “Locally Cartesian Closed Categories and Type Theory”, Mathematical Proceedings of the Cambridge Mathematical Society, 95, no. 1, 33–48.

184

Shapiro, S., 2005, “Categories, Structures and the Frege-Hilbert Controversy: the Status of Metamathematics”, Philosophia Mathematica, 13, 1, 61–77.

185

Sorkin, R.D.: 1991, Finitary substitute for continuous topology, Int. J. Theor. Phys. 30 No. 7.: 923–947.

186

Smolin, L.: 2001, Three Roads to Quantum Gravity, Basic Books: New York.

187

Spanier, E. H.: 1966, Algebraic Topology, McGraw Hill: New York.

188

Spencer–Brown, G.: 1969, Laws of Form, George Allen and Unwin, London.

189

Stapp, H.: 1993, Mind, Matter and Quantum Mechanics, Springer Verlag: Berlin–Heidelberg–New York.

190

Stewart, I. and Golubitsky, M. : 1993. “Fearful Symmetry: Is God a Geometer?”, Blackwell: Oxford, UK.

191

Szabo, R. J.: 2003, Quantum field theory on non-commutative spaces, Phys. Rep. 378: 207–209.

192

Tattersall, I. and J. Schwartz: 2000, Extinct Humans. Westview Press, Boulder, Colorado and Cumnor Hill Oxford. ISBN 0-8133-3482-9 (hc)

193

Thom, R.: 1980, Modèles mathématiques de la morphogénèse, Paris, Bourgeois.

194

Thompson, W. D'Arcy: 1994, On Growth and Form., Dover Publications, Inc: New York.

195

uring, A.M.: 1952, The Chemical Basis of Morphogenesis, Philosophical Trans. of the the Royal Soc.(B) 257:37-72.

196

Taylor, P., 1996, “Intuitionistic sets and Ordinals.”, Journal of Symbolic Logic, 61, 705–744.

197

Taylor, P., 1999, “Practical Foundations of Mathematics.”, Cambridge: Cambridge University Press.

198

Tierney, M., 1972, “Sheaf Theory and the Continuum Hypothesis”, in Toposes, Algebraic Geometry and Logic,

199

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.

200

Van der Hoeven, G. & Moerdijk, I., 1984a, “Sheaf Models for Choice Sequences”, Annals of Pure and Applied Logic, 27, no. 1, 63–107.

201

Van der Hoeven, G. & Moerdijk, I., 1984b, “On Choice Sequences determined by Spreads”, Journal of Symbolic Logic, 49, no. 3, 908–916.

202

Várilly, J. C.: 1997, An introduction to noncommutative geometry ( ).

203

von Neumann, J.: 1932, Mathematische Grundlagen der Quantenmechanik, Springer: Berlin.

204

Wallace, R.: 2005, Consciousness : A Mathematical Treatment of the Global Neuronal Workspace, Springer: Berlin.

205

Weinstein, A.: 1996, Groupoids : unifying internal and external symmetry, Notices of the Amer. Math. Soc. 43: 744–752.

206

Wess J. and J. Bagger: 1983, Supersymmetry and Supergravity, Princeton University Press: Princeton, NJ.

207

Weinberg, S.: 1995, The Quantum Theory of Fields vols. 1 to 3, Cambridge Univ. Press.

208

Wheeler, J. and W. Zurek: 1983, Quantum Theory and Measurement, Princeton University Press: Princeton, NJ.

209

Whitehead, J. H. C.: 1941, On adding relations to homotopy groups, Annals of Math. 42 (2): 409–428.

210

Wiener, N.: 1950, The Human Use of Human Beings: Cybernetics and Society. Free Association Books: London, 1989 edn.

211

Woit, P.: 2006, Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Laws, Jonathan Cape.

212

Wood, R.J., 2004, Ordered Sets via Adjunctions, Categorical Foundations, M. C. Pedicchio & W. Tholen, eds., Cambridge: Cambridge University Press.