From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2850 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: weak double categories? Date: Fri, 28 Oct 2005 10:10:55 +0100 (BST) Message-ID: References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=ISO-8859-1; FORMAT=flowed Content-Transfer-Encoding: QUOTED-PRINTABLE X-Trace: ger.gmane.org 1241018942 6077 80.91.229.2 (29 Apr 2009 15:29:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:02 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Fri Oct 28 16:28:02 2005 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Oct 2005 16:28:02 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EVZqc-0001US-1u for categories-list@mta.ca; Fri, 28 Oct 2005 16:25:06 -0300 In-Reply-To: Content-ID: Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 25 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:2850 Archived-At: --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=20 is the first person to do this, in his thesis.=20 Grandis and Par=E9 are the only people to have=20 developed extensively aspects of their theory ([1]=20 & [2]). Tom Leinster mentions them in passing (in=20 [3] for example) -- they are the `representable'=20 fc-multicategories, standing in the same relation=20 to them as monoidal categories do to plain=20 multicategories. On my website [4] is my thesis "Polycategories"=20 which contains a fair bit more on weak double=20 categories, both further aspects of their theory=20 and some applications; for those of a terser=20 inclination, the edited highlights can be found in=20 the two preprints "Double clubs" and=20 "Polycategories via pseudo-distributive laws" on=20 the same page. Richard Garner ----- [1] Marco Grandis & Robert Par=E9 Limits in double categories Cah. Topol. G=E9om. Diff=E9r. Cat=E9g. 40 (1999), no. 3, 162--220; MR171677= 9 (2000i:18007) [2] Marco Grandis & Robert Par=E9 Adjoints for double categories Cah. Topol. G=E9om. Diff=E9r. Cat=E9g. 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