Re: weak double categories?

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

--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