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Elementary observations on 2-categorical limits

  • G.M. Kelly (a1)
Abstract

With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important 2-categories fail to be complete but do admit a wide class of limits. Accordingly, we introduce a variety of particular 2-categorical limits of practical importance, and show that certain of these suffice for the existence of indexed lax- and pseudo-limits. Other important 2-categories fail to admit even pseudo-limits, but do admit the weaker bilimits; we end by discussing these.

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References
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[1]Auderset C., ‘Adjonctions et monades au niveau des 2-catégories’, Cahiers de Topologie et Géom. Diff. 15 (1974), 320.
[2]Bénabou J., ‘Introduction to bicategories’, in Lecture Notes in Math. 47, pp. 177 (Springer-Verlag, Berlin, Heidelberg, New York, 1967).
[3]Bird G.J., Limits of Locally-presentable Categories (Ph.D thesis, Univ. of Sydney, 1984).
[4]Bird G.J., Kelly G.M., Power A.J. and Street R., ‘Flexible limits for 2-categories’, J. Pure Appl. Algebra (to appear).
[5]Blackwell R., Kelly G.M. and Power J., ‘Two-dimensional monad theory’, J. Pure Appl Algebra (to appear).
[6]Borceux F. and Kelly G.M., ‘A notion of limit for enriched categories’, Bull. Austral. Math. Soc. 12 (1975), 4972.
[7]Gray J.W., ‘Formal Category Theory: Adjointness for 2-Categories’: Lecture Notes in Math. 391 (Springer-Verlag, Berlin, Heidelberg, New York).
[8]Kelly G.M., ‘On clubs and doctrines’, in Lecture Notes in Math. 420 (Springer-Verlag, Berlin, Heidelberg, New York, 1974). pp. 181256.
[9]Kelly G.M., Basic Concepts of Enriched Category Theory (London Math Soc. Lecture Notes Series 64, Cambridge Univ. Press, 1982).
[10]Kelly G.M., ‘Structures defined by finite limits in the enriched context I’, Cahiers de Topologie et Géom. Diff 23 (1982), 342.
[11]Kelly G.M., ‘Equivalences in 2-categories, birepresentations, and biadjoints’, (in preparation).
[12]Kelly G.M. and Street R., ‘Review of the elements of 2-categories’, in Lecture Notes in Math. 420 (Springer-Verlag, Berlin, Heidelberg, New York, 1974). pp. 75103.
[13]Street R., ‘Elementary Cosmoi I’, in Lecture Notes in Math 420 (Springer-Verlag, Berlin, Heidelberg, New York, 1974). pp. 134180.
[14]Street R., ‘Limits indexed by category-valued 2-functor’, J. Pure Appl. Algebra 8 (1976), 149181.
[15]Street R., ‘Fibrations in bicategories’, Cahiers de Topologie et Géom. Diff. 21 (1980), 111160.
[16]Street R., ‘Corrigendum to “Fibrations in bicategories”‘, Cahiers de Topologie et Géom. Diff. Catégoriques 28 (1987), 5356.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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