Advertisement

Applied Categorical Structures

, Volume 11, Issue 3, pp 219–227 | Cite as

Functorial Calculus in Monoidal Bicategories

  • Ross Street
Article

Abstract

The definition and calculus of extraordinary natural transformations is extended to a context internal to any autonomous monoidal bicategory. The original calculus is recaptured from the geometry of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules.

enriched category dual dinatural transformation Gray monoid 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bénabou, J.: Introduction to bicategories, Lecture Notes in Math. 47, Springer, Berlin, 1967, pp. 1–77.Google Scholar
  2. 2.
    Baez, J. and Dolan, J.: From finite sets to Feynman diagrams, in B. Engquist and W. Schmid (eds.), Mathematics Unlimited – 2001 and Beyond, Springer-Verlag, Berlin, to appear.Google Scholar
  3. 3.
    Day, B. J.: On closed categories of functors, Lecture Notes in Math. 137, Springer, Berlin, 1970, pp. 1–38.Google Scholar
  4. 4.
    Day, B. J.: Note on compact closed categories, Australian Mathematical Society Journal Series A 24 (1977), 309–311.Google Scholar
  5. 5.
    Day, B. J. and Street, R.: Monoidal bicategories and Hopf algebroids, Advances in Mathematics 129 (1997), 99–157.Google Scholar
  6. 6.
    Dubuc, E. and Street, R.: Dinatural transformations, Lecture Notes in Math. 137, Springer, Berlin, 1970, pp. 126–137.Google Scholar
  7. 7.
    Eilenberg, S. and Mac Lane, S.: Natural isomorphisms in group theory, Proceedings of the National Academy of Sciences of the U.S.A. 28 (1942), 537–543.Google Scholar
  8. 8.
    Eilenberg, S. and Mac Lane, S.: General theory of natural equivalences, Transactions of the American Mathematical Society 58 (1945), 231–294.Google Scholar
  9. 9.
    Eilenberg, S. and Kelly, G. M.: A generalization of the functorial calculus, Journal of Algebra 3 (1966), 366–375.Google Scholar
  10. 10.
    Eilenberg, S. and Kelly, G. M.: Closed categories, in Proceedings of the Conference on Categorical Algebra at La Jolla, Springer, 1966, pp. 421–562.Google Scholar
  11. 11.
    Gordon, R., Power, A. J. and Street, R.: Coherence for tricategories, Memoirs of the American Mathematical Society 117 #558 (1995) (ISBN 0-8218-0344-1).Google Scholar
  12. 12.
    Gray, J. W.: Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Math. 391, Springer, Berlin, 1974.Google Scholar
  13. 13.
    Gray, J. W.: 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
  14. 14.
    Joyal, A. and Street, R.: Braided tensor categories, Advances in Mathematics 102 (1993), 20–78.Google Scholar
  15. 15.
    Joyal, A. and Street, R.: The geometry of tensor calculus I, Advances in Mathematics 88 (1991), 55–112.Google Scholar
  16. 16.
    Kelly, G. M.: Tensor products in categories, Journal of Algebra 2 (1965), 15–37.Google Scholar
  17. 17.
    Kelly, G. M. and Laplaza, M. L.: Coherence for compact closed categories, Journal of Pure and Applied Algebra 19 (1980), 193–213.Google Scholar
  18. 18.
    Kelly, G. M. and Street, R.: Review of the elements of 2-categories, Lecture Notes in Math. 420, Springer, Berlin, 1974, pp. 75–103.Google Scholar
  19. 19.
    Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Math. 5, Springer-Verlag, 1971.Google Scholar
  20. 20.
    McIntyre, M. and Trimble, T.: The geometry of Gray monoids, Preprint.Google Scholar
  21. 21.
    Street, R. and Verity, D.: Low-dimensional topology and higher-order categories, in Proceedings of CT95, Halifax, July 9–15, 1995. http://www.mta.ca/~cat-dist/ct95.htmlGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Ross Street
    • 1
  1. 1.Centre of Australian Category TheoryMacquarie UniversityAustralia

Personalised recommendations