Dear John,
The definition of weak (or pseudo) double category is what you would expect, once you know the strict case and the definition of bicategory; think of a "pseudocategory object in Cat". It has probably been in the categorical folklore since quite a while.
You can find it (including the machinery of lax double functors, horizontal and vertical transformations, etc.) in two papers in Cahiers, where part of the general theory of weak double categories has been developed
M. Grandis - R. Pare, Limits in double categories, Cah. Topol. Geom. Diff. Categ. 40 (1999), 162-220,
- -, Adjoints for double categories, Cah. Topol. Geom. Diff. Categ. 45 (2004), 193-240.
Other papers on (weak) double categories, many of them by Bob Pare et al., are referred to in the articles above.
In book form
Tom Leinster, Higher operads, higher categories, Cambridge Un. Press 2004,
has Section 5.2, devoted to weak double categories.
Best regards
Marco
Marco Grandis
Dipartimento di Matematica
Universitą di Genova
Via Dodecaneso, 35
16146 Genova
Italy
tel: +39 010 353 6805
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