The definition of weak (or pseudo) double category is what you would =20
expect, once you know the strict case and the definition of =20
bicategory; think of a "pseudocategory object in Cat". It has =20
probably been in the categorical folklore since quite a while.

You can find it (including the machinery of lax double functors, =20
horizontal and vertical transformations, etc.) in two papers in =20
Cahiers, where part of the general theory of weak double categories =20
has been developed

    M. Grandis - R. Pare, Limits in double categories, Cah. Topol. =20
Geom. Diff. Categ. 40 (1999), 162-220,

    - -, Adjoints for double categories,  Cah. Topol. Geom. Diff. =20
Categ. 45 (2004), 193-240.

Other papers on (weak) double categories, many of them by Bob Pare et =20=

al., are referred to in the articles above.
In book form

    Tom Leinster, Higher operads, higher categories, Cambridge Un. =20
Press 2004,

has Section 5.2, devoted to weak double categories.

Best regards

Marco

Marco Grandis
Dipartimento di Matematica
Universit=E0 di Genova
Via Dodecaneso, 35
16146 Genova
Italy

e-mail: grandis@dima.unige.it
tel: +39 010 353 6805
http://www.dima.unige.it/~grandis/

On 26 Oct 2005, at 22:08, John Baez wrote:

> Dear Categorists -
>
> If you weaken the notion of 2-category you get the notion of
> bicategory.  Has anyone tried to correspondingly weaken the
> notion of double category, so that a bicategory is a special
> sort of "weak double category" in analogy to the ways in which
> a 2-category is a special sort of double category?  Did anyone
> succeed?
>
> Best,
> jb
>
>
>
>
>
>