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
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
[Note from moderator: message resent, may have been transmitted incorrectly.] Dominic Verity's PhD thesis did that (amongst other things) for some very good reasons. ---Ross On 27/10/2005, at 6:08 AM, 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?
Dominic Verity's PhD thesis did that (amongst other things) for some very good reasons. ---Ross On 27/10/2005, at 6:08 AM, 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?
--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
Dear Categorists - Thanks for all the helpful replies! Tom Leinster wrote:
I guess you're after weakening in both directions.
Yes, I need weakening in both directions for my particular application (to quantum gravity, in fact). So, I need to get ahold of Dominic Verity's thesis. Best, jb
participants (5)
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John Baez -
Marco Grandis -
Richard Garner -
Ross Street -
Tom Leinster