--On 26 October 2005 13:08 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? Did anyone succeed?
Yes, this has been done; I believe Dom Verity is the first person to do this, in his thesis. Grandis and Paré are the only people to have developed extensively aspects of their theory ([1] & [2]). Tom Leinster mentions them in passing (in [3] for example) -- they are the `representable' fc-multicategories, standing in the same relation to them as monoidal categories do to plain multicategories. On my website [4] is my thesis "Polycategories" which contains a fair bit more on weak double categories, both further aspects of their theory and some applications; for those of a terser inclination, the edited highlights can be found in the two preprints "Double clubs" and "Polycategories via pseudo-distributive laws" on the same page. Richard Garner ----- [1] Marco Grandis & Robert Paré Limits in double categories Cah. Topol. Géom. Différ. Catég. 40 (1999), no. 3, 162--220; MR1716779 (2000i:18007) [2] Marco Grandis & Robert Paré Adjoints for double categories Cah. Topol. Géom. Différ. Catég. 45 (2004), no. 3, 193--240. [3] Tom Leinster Higher operads, higher categories http://arxiv.org/abs/math.CT/0305049 [4] Richard Garner http://www.dpmms.cam.ac.uk/~rhgg2