Adjoints in bicategories
I'm looking for definitions of (the weak 2-dimensional analogues of 1-) products and coproducts for bicategories, and also adjoints. In his 1967 article "Introduction to Bicategories, Part I" Benabou promises to treat biadjoints in a sequel, but I gather this was never published. Gray treats a notion of "quasi-adjointness" in "Formal Category Theory"; is this accepted as the "right" generalization? Pointers to definitions of these concepts in one of the approaches to weak n-categories would be welcome as well. -- Frank Atanassow, Dept. of Computer Science, Utrecht University Padualaan 14, PO Box 80.089, 3508 TB Utrecht, Netherlands Tel +31 (030) 253-1012, Fax +31 (030) 251-3791
In reply to Frank Atanassow's question: Here is a list of a few references I have on hand that are related or somewhat related to your question. This list is not intended to be complete and I hope others will add more references and/or details. John MacDonald, Department of Mathematics University of British Columbia Vancouver, B.C., Canada V5K 1N4 [1] R. Blackwell, G.M.Kelly, J.Power, Two-Dimensional Monad Theory, Sydney Category Seminar Reports 1987. [2] M.C.Bunge, Coherent Extensions and Relational Algebras, Trans. Amer. Math. Soc.197(1974), 355-390. [3] J.W.Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Mathematics 391, Springer-Verlag 1974. [4] C.B.Jay, Local Adjunctions, Journal of Pure and Applied Algebra 53(1988), 227-238 [5] G.M.Kelly, Elementary Observations on 2-Categorical Limits, Bull. Austral. Math. Soc. 39(1989), 301-317 [6] G.M.Kelly, R.H.Street, Review of the Elements of 2-Categories, Lecture Notes in Mathematics 420, Springer-Verlag 1974, 75-109. [7] J.L.MacDonald, A.Stone, Soft Adjunction between 2-Categories, Journal of Pure and Applied Algebra 60(1989), 155-203. [8] R.H.Street, The formal Theory of Monads, Journal of Pure and Applied Algebra 2(1972),149-168. At 02:45 PM 6/22/00 +0200, you wrote:
I'm looking for definitions of (the weak 2-dimensional analogues of 1-) products and coproducts for bicategories, and also adjoints. In his 1967 article "Introduction to Bicategories, Part I" Benabou promises to treat biadjoints in a sequel, but I gather this was never published. Gray treats a notion of "quasi-adjointness" in "Formal Category Theory"; is this accepted as the "right" generalization?
Pointers to definitions of these concepts in one of the approaches to weak n-categories would be welcome as well.
-- Frank Atanassow, Dept. of Computer Science, Utrecht University Padualaan 14, PO Box 80.089, 3508 TB Utrecht, Netherlands Tel +31 (030) 253-1012, Fax +31 (030) 251-3791
Frank Atanassow wrote:
I'm looking for definitions of (the weak 2-dimensional analogues of 1-) products and coproducts for bicategories, and also adjoints. In his 1967 article "Introduction to Bicategories, Part I" Benabou promises to treat biadjoints in a sequel, but I gather this was never published. Gray treats a notion of "quasi-adjointness" in "Formal Category Theory"; is this accepted as the "right" generalization?
Pointers to definitions of these concepts in one of the approaches to weak n-categories would be welcome as well.
Relevant references: Kelly, G. M. Elementary observations on $2$-categorical limits. Bull. Austral. Math. Soc. 39 (1989), no. 2, 301--317. Power, A. J. Coherence for bicategories with finite bilimits. I. Categories in computer science and logic (Boulder, CO, 1987),341--347, Contemp. Math., 92, Amer. Math. Soc., Providence, RI, 1989. Betti, Renato; Power, A. John On local adjointness of distributive bicategories. Boll. Un. Mat. Ital. B (7) 2 (1988), no. 4, 931--947. Bird, G. J.; Kelly, G. M.; Power, A. J.; Street, R. H. Flexible limits for $2$-categories. J. Pure Appl. Algebra 61 (1989), no. 1, 1--27.
participants (3)
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Claudio Hermida -
Frank Atanassow -
John MacDonald