On Wed, 2005-10-26 at 13:08 -0700, John Baez wrote:
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?
At least three parties have done this: - Bob Pare and collaborators - at least one Australian of the Sean Carmody/Dominic Verity/Steve Lack generation (calling them something like "double bicategories") - me (section 5.2 of book). There's the question of whether you weaken in just one direction or in both. I believe that parties 1 and 3 weaken in just one direction. But you write
in analogy to the ways [*plural!*] in which a 2-category is a special sort of double category
so I guess you're after weakening in both directions. Tom