[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Functorial Calculus in Monoidal Bicategories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bénabou, J.: Introduction to bicategories, Lecture Notes in Math. 47, Springer, Berlin, 1967, pp. 1–77.

    Google Scholar 

  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.

  3. Day, B. J.: On closed categories of functors, Lecture Notes in Math. 137, Springer, Berlin, 1970, pp. 1–38.

    Google Scholar 

  4. Day, B. J.: Note on compact closed categories, Australian Mathematical Society Journal Series A 24 (1977), 309–311.

    Google Scholar 

  5. Day, B. J. and Street, R.: Monoidal bicategories and Hopf algebroids, Advances in Mathematics 129 (1997), 99–157.

    Google Scholar 

  6. Dubuc, E. and Street, R.: Dinatural transformations, Lecture Notes in Math. 137, Springer, Berlin, 1970, pp. 126–137.

    Google Scholar 

  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. Eilenberg, S. and Mac Lane, S.: General theory of natural equivalences, Transactions of the American Mathematical Society 58 (1945), 231–294.

    Google Scholar 

  9. Eilenberg, S. and Kelly, G. M.: A generalization of the functorial calculus, Journal of Algebra 3 (1966), 366–375.

    Google Scholar 

  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.

  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).

  12. Gray, J. W.: Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Math. 391, Springer, Berlin, 1974.

    Google Scholar 

  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. Joyal, A. and Street, R.: Braided tensor categories, Advances in Mathematics 102 (1993), 20–78.

    Google Scholar 

  15. Joyal, A. and Street, R.: The geometry of tensor calculus I, Advances in Mathematics 88 (1991), 55–112.

    Google Scholar 

  16. Kelly, G. M.: Tensor products in categories, Journal of Algebra 2 (1965), 15–37.

    Google Scholar 

  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. 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. Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Math. 5, Springer-Verlag, 1971.

  20. McIntyre, M. and Trimble, T.: The geometry of Gray monoids, Preprint.

  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.html

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Street, R. Functorial Calculus in Monoidal Bicategories. Applied Categorical Structures 11, 219–227 (2003). https://doi.org/10.1023/A:1024247613677

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024247613677

Navigation