Skip to main content
Log in

Formal aspects of Gray’s tensor products of 2-categories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

The category of small 2-categories has two monoidal structures due to John Gray: one biclosed and one closed. We propose a formalisation of the construction of the right internal and internal homs of these monoidal structures.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Batanin, M.A.: Coherent categories with respect to monads and coherent prohomotopy theory. Cah. Topol. Géom. Différ. Catég. 34(4), 279–304 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Batanin, M. A.: Homotopy coherent category theory and A  ∞ -structures in monoidal categories. J. Pure Appl. Algebra 123(1–3), 67–103 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourn, D.: Natural anadeses and catadeses. Cah. Topol. Géom. Différ. Catég. 14(4), 371–415, 480 (1973)

    MathSciNet  MATH  Google Scholar 

  4. Cordier, J.M., Porter, T.: Homotopy coherent category theory. Trans. Am. Math. Soc. 349(1), 1–54 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Day, B.: On closed categories of functors. In: Reports of the Midwest Category Seminar, IV, vol. 137, pp. 1–38. Lecture Notes in Mathematics (1970)

  6. Foltz, F., Lair, C., Kelly, G.M.: Algebraic categories with few monoidal biclosed structures or none. J. Pure Appl. Algebra 17(2), 171–177 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gray, J.W.: Formal category theory: adjointness for 2-categories. In: Lecture Notes in Mathematics, vol. 391, pp. xii+282. Springer, Berlin-New York (1974)

    Google Scholar 

  8. Gray, J.W.: Coherence for the Tensor Product of 2-Categories, and Braid Groups. Algebra, Topology, and Category Theory (a Collection of Papers in Honor of Samuel Eilenberg), pp. 63–76. Academic Press, New York (1976)

    Google Scholar 

  9. Gray, J.W.: Closed categories, lax limits and homotopy limits. J. Pure Appl. Algebra 19, 127–158 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Im, G.B., Kelly, G.M.: A universal property of the convolution monoidal structure. J. Pure Appl. Algebra 43(1), 75–88 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Janelidze, G., Kelly, G.M.: A note on actions of a monoidal category. CT2000 Conference (Como). Theory Appl. Categ. 9, 61–91 (2001/2002)

    MathSciNet  Google Scholar 

  12. Kelly, G.M.: Doctrinal adjunction. In: Category Seminar (Proc. Sem., Sydney, 1972/1973), pp. 257–280. Lecture Notes in Math., vol. 420. Springer, Berlin (1974)

    Chapter  Google Scholar 

  13. Mac Lane, S.: Categories for the working mathematician, 2nd edn. Graduate Texts in Mathematics, vol. 5, pp. xii+314. Springer, New York (1998)

    MATH  Google Scholar 

  14. McClure, J.E., Smith, J.H.: Cosimplicial objects and little n-cubes. I. Am. J. Math. 126(5), 1109–1153 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stanculescu, A.E.: Formal aspects of Gray’s tensor products of 2-categories II (in preparation)

  16. Warren, M.A.: A characterization of representable intervals. arXiv:0903.3743 [math.CT] (2009). March 2009

  17. Wolff, H.: V-cat and V-graph. J. Pure Appl. Algebra 4, 123–135 (1974)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexandru Emil Stanculescu.

Additional information

Research supported by the Ministry of Education of the Czech Republic under grant LC505.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stanculescu, A.E. Formal aspects of Gray’s tensor products of 2-categories. Appl Categor Struct 21, 781–800 (2013). https://doi.org/10.1007/s10485-012-9282-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-012-9282-7

Keywords

Mathematics Subject Classifications (2010)

Navigation