Adjoints in bicategories

Adjoints in bicategories

[Frank Atanassow]

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

Re: Adjoints in bicategories

[John MacDonald]

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
>
>
>
>

Re: Adjoints in bicategories

[Claudio Hermida]

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.