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An Australian Conspectus of Higher Categories

  • Ross StreetEmail author
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 152)

Abstract

Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higher-dimensional work.

Keywords

Category Theory Parity Complex Monoidal Structure Enrich Category Baxter Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    I. Aitchison, String diagrams for non-abelian cocycle conditions, handwritten notes, talk presented at Louvain-la-neuve, Belgium, 1987.Google Scholar
  2. [2]
    J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995), 6073–6105.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    J. Baez and J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes, Advances in Math. 135 (1998), 145–206.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Baez and L. Langford, Higher-dimensional algebra IV: 2-tangles, Advances in Math. 180 (2003), 705–764.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Baez and M. Neuchl, Higher-dimensional algebra I: braided monoidal 2-categories, Advances in Math. 121 (1996), 196–244.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    C. Balteanu, Z. Fierderowicz, R. Schwaenzl, and R. Vogt, Iterated monoidal categories, Advances in Math. 176 (2003), 277–349.zbMATHCrossRefGoogle Scholar
  7. [7]
    M. Barr, Relational algebras, Lecture Notes in Math. 137, Springer, Berlin, 1970, 39–55.Google Scholar
  8. [8]
    M. Batanin, Monoidal globular categories as natural environment for the theory of weak n-categories, Advances in Math. 136 (1998), 39–103.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Batanin, Computads for finitary monads on globular sets, in Higher Category Theory, eds. E. Getzler and M. Kapranov, Contemp. Math. 230, AMS, Providence, Rhode Island, 1998, pp. 1–36.Google Scholar
  10. [10]
    M. Batanin and R. Street, The universal property of the multitude of trees, J. Pure Appl. Algebra 154 (2000), 3–13.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    J. Beck, Triples, algebras and cohomology, Reprints in Theory Appl. Cat. 2 (2003), 1–59.Google Scholar
  12. [12]
    J. Bénabou, Introduction to bicategories, Lecture Notes in Math. 47, Springer, Berlin, 1967, pp. 1–77.Google Scholar
  13. [13]
    J. Bénabou, Les distributeurs, Univ. Catholique de Louvain, Séminaires de Math. Pure, Rapport No. 33 (1973).Google Scholar
  14. [14]
    C. Berger, Double loop spaces, braided monoidal categories and algebraic 3-type of space, in Higher Homotopy Structures in Topology and Mathematical Physics, ed. J. McCleary, Contemp. Math. 227, AMS, Providence, Rhode Island, 1999, pp. 49–66.Google Scholar
  15. [15]
    R. Betti, A. Carboni, R. Street, and R. Walters, Variation through enrichment, J. Pure Appl. Algebra 29 (1983), 109–127.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    G.J. Bird, G.M. Kelly, A.J. Power, and R. Street, Flexible limits for 2-categories, J. Pure Appl. Algebra 61 (1989), 1–27.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    R. Blackwell, G.M. Kelly, and A.J. Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989), 1–41.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    F. Borceux and G.M. Kelly, A notion of limit for enriched categories, Bull. Austral. Math. Soc. 12 (1975), 49–72.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    R. Brown, Higher dimensional group theory, in Low Dimensional Topology, London Math. Soc. Lecture Note Series 48 (1982), pp. 215–238.Google Scholar
  20. [20]
    R. Brown and P.J. Higgins, The equivalence of crossed complexes and 1-groupoids, Cah. Top. Géom. Diff. Cat. 22 (1981), 371–386.zbMATHMathSciNetGoogle Scholar
  21. [21]
    A. Carboni, G.M. Kelly, and R.J. Wood, A 2-categorical approach to change of base and geometric morphisms. I., Cah. Topologie Géom. Diff. Cat. 32 (1991), 47–95.zbMATHMathSciNetGoogle Scholar
  22. [22]
    A. Carboni, G.M. Kelly, D. Verity, and R.J. Wood, A 2-categorical approach to change of base and geometric morphisms. II., Theory Appl. Cat. 4 (1998), 82–136.zbMATHMathSciNetGoogle Scholar
  23. [23]
    A. Carboni, S. Johnson, R. Street, and D. Verity, Modulated bicategories, J. Pure Appl. Algebra 94 (1994), 229–282.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    A. Carboni, S. Kasangian, and R. Walter, An axiomatics for bicategories of modules, J. Pure Appl. Algebra 45 (1987), 127–141.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    A. Carboni and R. Walters, Cartesian bicategories I, J. Pure Appl. Algebra 49 (1987), 11–32.zbMATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    S.M. Carmody, Cobordism Categories, Ph.D. Thesis, University of Cambridge, 1995.Google Scholar
  27. [27]
    L. Crane and D. Yetter, A categorical construction of 4D topological quantum field theories, in Quantum Topology, eds. L. Kauffman and R. Baadhio, World Scientific Press, 1993, pp. 131–138.Google Scholar
  28. [28]
    S. Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Advances in Math. 136 (1998), 183–223.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    S. Crans, A tensor product for Gray-categories, Theory Appl. Cat. 5 (1999), 12–69.zbMATHMathSciNetGoogle Scholar
  30. [30]
    S. Crans, On braidings, syllepses, and symmetries, Cah. Top. Géom. Diff. Cat. 41 (2000), 2–74 & 156.zbMATHMathSciNetGoogle Scholar
  31. [31]
    M. Dakin, Kan Complexes and Multiple Groupoid Structures, Ph.D. Thesis, University of Wales, Bangor, 1977.Google Scholar
  32. [32]
    B. Day, On closed categories of functors, Lecture Notes in Math. 137, Springer, Berlin, 1970, pp. 1–38.Google Scholar
  33. [33]
    B. Day and G.M. Kelly, Enriched functor categories, Reports of the Midwest Category Seminar, III, Springer, Berlin, 1969, pp. 178–191.CrossRefGoogle Scholar
  34. [34]
    B. Day, P. McCrudden, and R. Street, Dualizations and antipodes, Applied Categorical Structures 11 (2003), 229–260.MathSciNetCrossRefGoogle Scholar
  35. [35]
    B. Day and R. Street, Monoidal bicategories and Hopf algebroids, Advances in Math. 129 (1997), 99–157.zbMATHMathSciNetCrossRefGoogle Scholar
  36. [36]
    B. Day and R. Street, Lax monoids, pseudo-operads, and convolution, in Diagrammatic Morphisms and Applications, eds. D. Radford, F. Souza, and D. Yetter, Contemp. Math. 318, AMS, Providence, Rhode Island, 2003, pp. 75–96.Google Scholar
  37. [37]
    B. Day and R. Street, Abstract substitution in enriched categories, J. Pure Appl. Algebra 179 (2003), 49–63.zbMATHMathSciNetCrossRefGoogle Scholar
  38. [38]
    B. Day and R. Street, Quantum categories, star autonomy, and quantum groupoids, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications 43, AMS, Providence, Rhode Island, 2004, pp. 193–231.Google Scholar
  39. [39]
    V.G. Drinfel’d, Quasi-Hopf algebras (Russian), Algebra i Analiz 1 (1989), 114–148; translation in Leningrad Math. J. 1 (1990), 1419–1457.MathSciNetGoogle Scholar
  40. [40]
    J.W. Duskin, The Azumaya complex of a commutative ring, Lecture Notes in Math. 1348, Springer, Berlin, 1988, pp. 107–117.Google Scholar
  41. [41]
    J.W. Duskin, An outline of a theory of higher-dimensional descent, Bull. Soc. Math. Belg. Sér. A 41 (1989), 249–277.zbMATHMathSciNetGoogle Scholar
  42. [42]
    J.W. Duskin, A simplicial-matrix approach to higher dimensional category theory I: nerves of bicategories, Theory Appl. Cat. 9 (2002), 198–308.MathSciNetGoogle Scholar
  43. [43]
    J.W. Duskin, A simplicial-matrix approach to higher dimensional category theory II: bicategory morphisms and simplicial maps (incomplete draft 2001).Google Scholar
  44. [44]
    C. Ehresmann, Catégories et Structures, Dunod, Paris, 1965.zbMATHGoogle Scholar
  45. [45]
    S. Eilenberg and G.M. Kelly, A generalization of the functorial calculus, J. Algebra 3 (1966), 366–375.zbMATHMathSciNetCrossRefGoogle Scholar
  46. [46]
    S. Eilenberg and G.M. Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra at La Jolla, Springer, Berlin, 1966, pp. 421–562.Google Scholar
  47. [47]
    S. Eilenberg and J.C. Moore, Adjoint functors and triples, Illinois J. Math. 9 (1965), 381–398.zbMATHMathSciNetGoogle Scholar
  48. [48]
    S. Eilenberg and R. Street, Rewrite systems, algebraic structures, and higher-order categories (handwritten notes circa 1986, somewhat circulated).Google Scholar
  49. [49]
    J. Fischer, 2-categories and 2-knots, Duke Math. Journal 75 (1994), 493–526.zbMATHCrossRefGoogle Scholar
  50. [50]
    P. Freyd and R. Street, On the size of categories, Theory Appl. Cat. 1 (1995), 174–178.zbMATHMathSciNetGoogle Scholar
  51. [51]
    P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer, Berlin, 1967.zbMATHGoogle Scholar
  52. [52]
    J. Giraud, Cohomologie Non Abélienne, Springer, Berlin, 1971.zbMATHGoogle Scholar
  53. [53]
    R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris, 1964.Google Scholar
  54. [54]
    R. Gordon, A.J. Power, and R. Street, Coherence for Tricategories, Mem. Amer. Math. Soc. 117 (1995), No. 558.Google Scholar
  55. [55]
    J.W. Gray, Category-valued sheaves, Bull. Amer. Math. Soc. 68 (1962), 451–453.zbMATHMathSciNetCrossRefGoogle Scholar
  56. [56]
    J.W. Gray, Sheaves with values in a category, Topology 3 (1965), 1–18.zbMATHMathSciNetCrossRefGoogle Scholar
  57. [57]
    J.W. Gray, Fibred and cofibred categories, Proceedings of the Conference on Categorical Algebra at La Jolla, Springer, Berlin, 1966, pp. 21–83.Google Scholar
  58. [58]
    J.W. Gray, The categorical comprehension scheme, Category Theory, Homology Theory and their Applications, III, Springer, Berlin, 1969, pp. 242–312.CrossRefGoogle Scholar
  59. [59]
    J.W. Gray, The 2-adjointness of the fibred category construction, Symposia Mathematica IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 457–492.Google Scholar
  60. [60]
    J.W. Gray, Report on the meeting of the Midwest Category Seminar in Zürich, Lecture Notes in Math. 195, Springer, Berlin, 1971, 248–255.Google Scholar
  61. [61]
    J.W. Gray, Quasi-Kan extensions for 2-categories, Bull. Amer. Math. Soc. 80 (1974), 142–147.zbMATHMathSciNetCrossRefGoogle Scholar
  62. [62]
    J.W. Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Math. 391, Springer, Berlin, 1974.Google Scholar
  63. [63]
    J.W. Gray, Coherence for the tensor product of 2-categories, and braid groups, in Algebra, Topology, and Category Theory (a collection of papers in honour of Samuel Eilenberg), Academic Press, New York, 1976, pp. 63–76.Google Scholar
  64. [64]
    J.W. Gray, Fragments of the history of sheaf theory, Lecture Notes in Math. 753, Springer, Berlin, 1979, pp. 1–79.Google Scholar
  65. [65]
    J.W. Gray, The existence and construction of lax limits, Cah. Top. Géom. Diff. Cat. 21 (1980), 277–304.zbMATHGoogle Scholar
  66. [66]
    J.W. Gray, Closed categories, lax limits and homotopy limits, J. Pure Appl. Algebra 19 (1980), 127–158.zbMATHMathSciNetCrossRefGoogle Scholar
  67. [67]
    J.W. Gray, Enriched algebras, spectra and homotopy limits, Lecture Notes in Math. 962, Springer, Berlin, 1982, pp. 82–99.Google Scholar
  68. [68]
    J.W. Gray, The representation of limits, lax limits and homotopy limits as sections, Contemp. Math. 30, AMS, Providence, Rhode Island, 1984, pp. 63–83.Google Scholar
  69. [69]
    M. Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete 64, Springer, Berlin, 1972.Google Scholar
  70. [70]
    C. Hermida, M. Makkai, and J. Power, On weak higher dimensional categories (preprint 1997 at http://www.math.mcgill.ca/makkai/).
  71. [71]
    P.J. Hilton and S. Wylie, Homology Theory: An Introduction to Algebraic Topology, Cambridge University Press, Cambridge, 1960.zbMATHGoogle Scholar
  72. [72]
    M. Johnson, Pasting Diagrams in n-Categories with Applications to Coherence Theorems and Categories of Paths, Ph.D. Thesis, University of Sydney, Australia, 1987.Google Scholar
  73. [73]
    M. Johnson, The combinatorics of n-categorical pasting, J. Pure Appl. Algebra 62 (1989), 211–225.zbMATHMathSciNetCrossRefGoogle Scholar
  74. [74]
    M. Johnson and R. Walters, On the nerve of an n-category, Cah. Top. Géom. Diff. Cat. 28 (1987), 257–282.zbMATHMathSciNetGoogle Scholar
  75. [75]
    A. Joyal, Disks, duality and Θ-categories, preprint and talk at the AMS Meeting in Montréal (September 1997).Google Scholar
  76. [76]
    A. Joyal and R. Street, The geometry of tensor calculus I, Advances in Math. 88 (1991), 55–112.zbMATHMathSciNetCrossRefGoogle Scholar
  77. [77]
    A. Joyal and R. Street, Tortile Yang–Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), 43–51.zbMATHMathSciNetCrossRefGoogle Scholar
  78. [78]
    A. Joyal and R. Street, An introduction to Tannaka duality and quantum groups, Lecture Notes in Math. 1488, Springer, Berlin, 1991, pp. 411–492.Google Scholar
  79. [79]
    A. Joyal and R. Street, Pullbacks equivalent to pseudopullbacks, Cah. Top. Géom. Diff. Cat. 34 (1993), 153–156.zbMATHMathSciNetGoogle Scholar
  80. [80]
    A. Joyal and R. Street, Braided tensor categories, Advances in Math. 102 (1993), 20–78.zbMATHMathSciNetCrossRefGoogle Scholar
  81. [81]
    A. Joyal, R. Street, and D. Verity, Traced monoidal categories, Math. Proc. Cambridge Philos. Soc. 119 (1996), 447–468.zbMATHMathSciNetCrossRefGoogle Scholar
  82. [82]
    M.M. Kapranov and V.A. Voevodsky, Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (List of results), Cah. Topologie et Géom. Diff. Cat. 32 (1991), 11–27.zbMATHMathSciNetGoogle Scholar
  83. [83]
    M.M. Kapranov and V.A. Voevodsky, ∞-Groupoids and homotopy types, Cah. Top. Géom. Diff. Cat. 32 (1991), 29–46.zbMATHMathSciNetGoogle Scholar
  84. [84]
    M.M. Kapranov and V.A. Voevodsky, 2-Categories and Zamolodchikov tetrahedra equations, Proc. Symp. Pure Math. 56 (1994), 177–259.MathSciNetGoogle Scholar
  85. [85]
    M.M. Kapranov and V.A. Voevodsky, Braided monoidal 2-categories and Manin–Schechtman higher braid groups, J. Pure Appl. Algebra 92 (1994), 241–267.zbMATHMathSciNetCrossRefGoogle Scholar
  86. [86]
    C. Kassel, Quantum Groups, Springer, Berlin, 1995.zbMATHGoogle Scholar
  87. [87]
    G.M. Kelly, Observations on the Köunneth theorem, Proc. Cambridge Philos. Soc. 59 (1963), 575–587.zbMATHMathSciNetCrossRefGoogle Scholar
  88. [88]
    G.M. Kelly, Complete functors in homology. I. Chain maps and endomorphisms, Proc. Cambridge Philos. Soc. 60 (1964), 721–735.MathSciNetCrossRefGoogle Scholar
  89. [89]
    G.M. Kelly, Complete functors in homology. II. The exact homology sequence, Proc. Cambridge Philos. Soc. 60 (1964), 737–749.MathSciNetCrossRefGoogle Scholar
  90. [90]
    G.M. Kelly, On Mac Lane’s conditions for coherence of natural associativities, commutativities, etc., J. Algebra 1 (1964), 397–402.zbMATHMathSciNetCrossRefGoogle Scholar
  91. [91]
    G.M. Kelly, A lemma in homological algebra, Proc. Cambridge Philos. Soc. 61 (1965), 49–52.zbMATHMathSciNetCrossRefGoogle Scholar
  92. [92]
    G.M. Kelly, Tensor products in categories, J. Algebra 2 (1965), 15–37.zbMATHMathSciNetCrossRefGoogle Scholar
  93. [93]
    G.M. Kelly, Chain maps inducing zero homology maps, Proc. Cambridge Philos. Soc. 61 (1965), 847–854.zbMATHMathSciNetCrossRefGoogle Scholar
  94. [94]
    G.M. Kelly, Adjunction for enriched categories, in Reports of the Midwest Category Seminar, III, Lecture Notes in Math. 106, Springer, Berlin, 1969, pp. 166–177.CrossRefGoogle Scholar
  95. [95]
    G.M. Kelly, Many-variable functorial calculus. I, in Coherence in Categories, Lecture Notes in Math. 281, Springer, Berlin, 1972, pp. 66–105.CrossRefGoogle Scholar
  96. [96]
    G.M. Kelly, An abstract approach to coherence, in Coherence in Categories, Lecture Notes in Math. 281, Springer, Berlin, 1972, pp. 106–147.CrossRefGoogle Scholar
  97. [97]
    G.M. Kelly, A cut-elimination theorem, in Coherence in Categories, Lecture Notes in Math. 281, Springer, Berlin, 1972, pp. 196–213.CrossRefGoogle Scholar
  98. [98]
    G.M. Kelly, Doctrinal adjunction, in Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math. 420, Springer, Berlin, 1974, pp. 257–280.Google Scholar
  99. [99]
    G.M. Kelly, On clubs and doctrines, in Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math. 420, Springer, Berlin, 1974, pp. 181–256.Google Scholar
  100. [100]
    G.M. Kelly, Coherence theorems for lax algebras and for distributive laws, in: Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math. 420, Springer, Berlin, 1974, pp. 281–375.Google Scholar
  101. [101]
    G.M. Kelly, Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Notes Series 64, Cambridge University Press, Cambridge, 1982. Also in Reprints in Theory Appl. Cat. 10 (2005), 1–136.Google Scholar
  102. [102]
    G.M. Kelly, Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc. 39 (1989), 301–317.zbMATHMathSciNetCrossRefGoogle Scholar
  103. [103]
    G.M. Kelly, On clubs and data-type constructors, London Math. Soc. Lecture Note Ser. 177, Cambridge Univ. Press, Cambridge, 1992, pp. 163–190.Google Scholar
  104. [104]
    G.M. Kelly, A. Labella, V. Schmitt, and R. Street, Categories enriched on two sides, J. Pure Appl. Algebra 168 (2002), 53–98.zbMATHMathSciNetCrossRefGoogle Scholar
  105. [105]
    G.M. Kelly and M.L. Laplaza, Coherence for compact closed categories, J. Pure Appl. Algebra 19 (1980), 193–213.zbMATHMathSciNetCrossRefGoogle Scholar
  106. [106]
    G.M. Kelly and S. Mac Lane, Coherence in closed categories, J. Pure Appl. Algebra 1 (1971), 97–140.zbMATHMathSciNetCrossRefGoogle Scholar
  107. [107]
    G.M. Kelly and S. Mac Lane, Erratum: Coherence in closed categories, J. Pure Appl. Algebra 1 (1971), p. 219.MathSciNetCrossRefGoogle Scholar
  108. [108]
    G.M. Kelly and R. Street, eds., Abstracts of the Sydney Category Seminar 1972/3. (First edition with brown cover, U. of New South Wales; second edition with green cover, Macquarie U.)Google Scholar
  109. [109]
    G.M. Kelly and R. Street, Review of the elements of 2-categories, Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math. 420, Springer, Berlin, 1974, pp. 75–103.Google Scholar
  110. [110]
    V. Kharlamov and V. Turaev, On the definition of the 2-category of 2-knots, Amer. Math. Soc. Transl. 174 (1996), 205–221.MathSciNetGoogle Scholar
  111. [111]
    A. Kock, Monads for which structures are adjoint to units, rhus Univ. Preprint Series 35 (1972–73), 1–15.Google Scholar
  112. [112]
    A. Kock, Monads for which structures are adjoint to units, J. Pure Appl. Algebra 104 (1995), 41–59.zbMATHMathSciNetCrossRefGoogle Scholar
  113. [113]
    S. Lack and R. Street, The formal theory of monads II, J. Pure Appl. Algebra 175 (2002), 243–265.zbMATHMathSciNetCrossRefGoogle Scholar
  114. [114]
    L. Langford, 2-Tangles as a Free Braided Monoidal 2-Category with Duals, Ph.D. dissertation, U. of California at Riverside, 1997.Google Scholar
  115. [115]
    F.W. Lawvere, The category of categories as a foundation for mathematics, in Proceedings of the Conference on Categorical Algebra at La Jolla, Springer, Berlin, 1966, pp. 1–20.Google Scholar
  116. [116]
    F.W. Lawvere, Ordinal sums and equational doctrines, in Seminar on Triples and Categorical Homology Theory, Lecture Notes in Math. 80 (1969), 141–155.MathSciNetCrossRefGoogle Scholar
  117. [117]
    F.W. Lawvere, Metric spaces, generalised logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1974), 135–166. Also in Reprints in Theory Appl. Cat. 1 (2002), pp. 1–37.MathSciNetCrossRefGoogle Scholar
  118. [118]
    J.-L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (1982), 179–202.zbMATHMathSciNetCrossRefGoogle Scholar
  119. [119]
    M. Mackaay, Spherical 2-categories and 4-manifold invariants, Advances in Math. 143 (1999), 288–348.zbMATHMathSciNetCrossRefGoogle Scholar
  120. [120]
    S. Mac Lane, Possible programs for categorists, Lecture Notes in Math. 86, Springer, Berlin, 1969, pp. 123–131.Google Scholar
  121. [121]
    S. Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 1971.zbMATHGoogle Scholar
  122. [122]
    S. Mac Lane and R. Paré, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), 59–80.MathSciNetCrossRefGoogle Scholar
  123. [123]
    F. Marmolejo, Distributive laws for pseudomonads, Theory Appl. Cat. 5 (1999), 91–147.zbMATHMathSciNetGoogle Scholar
  124. [124]
    P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Math. 271, Springer, Berlin, 1972.zbMATHGoogle Scholar
  125. [125]
    M. McIntyre and T. Trimble, The geometry of Gray-categories, Advances in Math. (to appear).Google Scholar
  126. [126]
    A. Pitts, Applications of sup-lattice enriched category theory to sheaf theory, Proc. London Math. Soc. (3) 57 (1988), 433–480.zbMATHMathSciNetCrossRefGoogle Scholar
  127. [127]
    A.J. Power, An n-categorical pasting theorem, in Category Theory, Proceedings, Como 1990, eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lecture Notes in Math. 1488 Springer, Berlin, 1991, pp. 326–358.Google Scholar
  128. [128]
    A.J. Power, Why tricategories?, Inform. & Comput. 120 (1995), 251–262.zbMATHMathSciNetCrossRefGoogle Scholar
  129. [129]
    N. Yu. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1–26.zbMATHMathSciNetCrossRefGoogle Scholar
  130. [130]
    J.E. Roberts, Mathematical aspects of local cohomology, Algèbres d’Opérateurs et leurs Applications en Physique Mathématique (Proc. Colloq., Marseille, 1977), Colloq. Internat. CNRS 274, CNRS, Paris, 1979, pp. 321–332.Google Scholar
  131. [131]
    R.D. Rosebrugh And R.J. Wood, Proarrows and cofibrations, J. Pure Appl. Algebra 53 (1988), 271–296.zbMATHMathSciNetCrossRefGoogle Scholar
  132. [132]
    G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112.zbMATHCrossRefGoogle Scholar
  133. [133]
    M.C. Shum, Tortile Tensor Categories, Ph.D. Thesis, Macquarie University, 1989; J. Pure Appl. Algebra 93 (1994), 57–110.zbMATHMathSciNetCrossRefGoogle Scholar
  134. [134]
    R. Street, Homotopy Classification of Filtered Complexes, Ph.D. Thesis, University of Sydney, 1968.Google Scholar
  135. [135]
    R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), 149–168.zbMATHMathSciNetCrossRefGoogle Scholar
  136. [136]
    R. Street, Two constructions on lax functors, Cah. Top. Géom. Diff. Cat. 13 (1972), 217–264.zbMATHMathSciNetGoogle Scholar
  137. [137]
    R. Street, Homotopy classification of filtered complexes, J. Australian Math. Soc. 15 (1973), 298–318.zbMATHMathSciNetCrossRefGoogle Scholar
  138. [138]
    R. Street, Fibrations and Yoneda’s lemma in a 2-category, in Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math. 420, Springer, Berlin, 1974, pp. 104–133.Google Scholar
  139. [139]
    R. Street, Elementary cosmoi I, in Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math. 420, Springer, Berlin, 1974, pp. 134–180.Google Scholar
  140. [140]
    R. Street, Limits indexed by category-valued 2–functors, J. Pure Appl. Algebra 8 (1976), 149–181.zbMATHMathSciNetCrossRefGoogle Scholar
  141. [141]
    R. Street, Cosmoi of internal categories, Trans. Amer. Math. Soc. 258 (1980), 271–318.zbMATHMathSciNetCrossRefGoogle Scholar
  142. [142]
    R. Street, Fibrations in bicategories, Cah. Top. Géom. Diff. Cat. 21 (1980), 111–160.zbMATHMathSciNetGoogle Scholar
  143. [143]
    R. Street, Conspectus of variable categories, J. Pure Appl. Algebra 21 (1981), 307–338.zbMATHMathSciNetCrossRefGoogle Scholar
  144. [144]
    R. Street, Cauchy characterization of enriched categories, Rendiconti del Seminario Matematico e Fisico di Milano 51 (1981), 217–233. (See [164].)zbMATHMathSciNetCrossRefGoogle Scholar
  145. [145]
    R. Street, Two dimensional sheaf theory, J. Pure Appl. Algebra 23 (1982), 251–270.zbMATHMathSciNetCrossRefGoogle Scholar
  146. [146]
    R. Street, Characterization of bicategories of stacks, Lecture Notes in Math. 962 (1982), 282–291.MathSciNetCrossRefGoogle Scholar
  147. [147]
    R. Street, Enriched categories and cohomology, Quaestiones Math. 6 (1983), 265–283.zbMATHMathSciNetGoogle Scholar
  148. [148]
    R. Street, Absolute colimits in enriched categories, Cah. Top. Géom. Diff. Cat. 24 (1983), 377–379.zbMATHMathSciNetGoogle Scholar
  149. [149]
    R. Street, Homotopy classification by diagrams of interlocking sequences, Math. Colloquium University of Cape Town 13 (1984), 83–120.MathSciNetGoogle Scholar
  150. [150]
    R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), 283–335.zbMATHMathSciNetCrossRefGoogle Scholar
  151. [151]
    R. Street, Correction to Fibrations in bicategories, Cah. Top. Géom. Diff. Cat. 28 (1987), 53–56.zbMATHMathSciNetGoogle Scholar
  152. [152]
    R. Street, Fillers for nerves, Lecture Notes in Math. 1348, Springer, Berlin, 1988, pp. 337–341.Google Scholar
  153. [153]
    R. Street, Parity complexes, Cah. Top. Géom. Diff. Cat. 32 (1991), 315–343.zbMATHMathSciNetGoogle Scholar
  154. [154]
    R. Street, Categorical structures, in Handbook of Algebra, Volume 1, ed. M. Hazewinkel, Elsevier, Amsterdam, 1996, pp. 529–577.Google Scholar
  155. [155]
    R. Street, Higher categories, strings, cubes and simplex equations, Applied Categorical Structures 3 (1995), 29–77 & 303.zbMATHMathSciNetCrossRefGoogle Scholar
  156. [156]
    R. Street, Parity complexes: corrigenda, Cah. Top. Géom. Diff. Cat. 35 (1994), 359–361.zbMATHMathSciNetGoogle Scholar
  157. [157]
    R. Street, Low-dimensional topology and higher-order categories, Proceedings of CT95, Halifax, July 9–15 1995; http://www.maths.mq.edu.au/~street/LowDTop.pdf.
  158. [158]
    R. Street, The role of Michael Batanin’s monoidal globular categories, in Higher Category Theory, eds. E. Getzler and M. Kapranov, Contemp. Math. 230, AMS, Providence, Rhode Island, 1998, pp. 99–116.Google Scholar
  159. [159]
    R. Street, The petit topos of globular sets, J. Pure Appl. Algebra 154 (2000), 299–315.zbMATHMathSciNetCrossRefGoogle Scholar
  160. [160]
    R. Street, Functorial calculus in monoidal bicategories, Applied Categorical Structures 11 (2003), 219–227.MathSciNetCrossRefGoogle Scholar
  161. [161]
    R. Street, Weak omega-categories, in Diagrammatic Morphisms and Applications, eds. D. Radford, F. Souza and D. Yetter, Contemp. Math. 318, AMS, Providence, Rhode Island, 2003, pp. 207–213.Google Scholar
  162. [162]
    R. Street, Categorical and combinatorial aspects of descent theory, Applied Categorical Structures 12 (2004), 537–576.zbMATHMathSciNetCrossRefGoogle Scholar
  163. [163]
    R. Street, Frobenius monads and pseudomonoids, J. Math. Phys. 45(10) (October 2004), 3930–3948.zbMATHMathSciNetCrossRefGoogle Scholar
  164. [164]
    R. Street, Cauchy characterization of enriched categories, Reprints in Theory and Applications of Categories 4 (2004), 1–16. (See [144].)MathSciNetGoogle Scholar
  165. [165]
    R. Street and R.F.C. Walters, Yoneda structures on 2-categories, J. Algebra 50 (1978), 350–379.zbMATHMathSciNetCrossRefGoogle Scholar
  166. [166]
    Z. Tamsamani, Sur des notions de n-categorie et n-groupoide non-stricte via des ensembles multi-simpliciaux, K-Theory 16 (1999), 51–99.zbMATHMathSciNetCrossRefGoogle Scholar
  167. [167]
    T. Trimble, The definition of tetracategory (handwritten diagrams; August 1995).Google Scholar
  168. [168]
    V.G. Turaev, The Yang–Baxter equation and invariants of links, Invent. Math. 92 (1988), 527–553.zbMATHMathSciNetCrossRefGoogle Scholar
  169. [169]
    D. Verity, Complicial sets, Mem. Amer. Math. Soc. (to appear; arXiv:math/0410412v2).Google Scholar
  170. [170]
    R.F.C. Walters, Sheaves on sites as Cauchy-complete categories, J. Pure Appl. Algebra 24 (1982), 95–102.zbMATHMathSciNetCrossRefGoogle Scholar
  171. [171]
    H. Wolff, Cat and graph, J. Pure Appl. Algebra 4 (1974), 123–135.zbMATHMathSciNetCrossRefGoogle Scholar
  172. [172]
    V. Zöberlein, Doctrines on 2-categories, Math. Z. 148 (1976), 267–279. (Originally Doktrinen auf 2-Kategorien, manuscript, Math. Inst. Univ. Zürich, 1973.)zbMATHMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag New York 2010

Authors and Affiliations

  1. 1.Centre of Australian Category TheoryMacquarie UniversityMacquarieAustralia

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