From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2848 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: weak double categories? Date: Fri, 28 Oct 2005 13:32:04 +0200 Message-ID: <633CF610-8823-4F2D-BF2F-63911C9E13A9@dima.unige.it> References: NNTP-Posting-Host: main.gmane.org Content-Type: multipart/alternative; boundary=Apple-Mail-6--279414351 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018941 6064 80.91.229.2 (29 Apr 2009 15:29:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:01 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Oct 28 16:28:01 2005 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Oct 2005 16:28:01 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EVZrc-0001a8-ON for categories-list@mta.ca; Fri, 28 Oct 2005 16:26:08 -0300 Original-Content-Type: text/plain;charset=ISO-8859-1;delsp=yes;format=flowed Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 23 Original-Lines: 163 Xref: news.gmane.org gmane.science.mathematics.categories:2848 Archived-At: Dear John, The definition of weak (or pseudo) double category is what you would =20 expect, once you know the strict case and the definition of =20 bicategory; think of a "pseudocategory object in Cat". It has =20 probably been in the categorical folklore since quite a while. You can find it (including the machinery of lax double functors, =20 horizontal and vertical transformations, etc.) in two papers in =20 Cahiers, where part of the general theory of weak double categories =20 has been developed M. Grandis - R. Pare, Limits in double categories, Cah. Topol. =20 Geom. Diff. Categ. 40 (1999), 162-220, - -, Adjoints for double categories, Cah. Topol. Geom. Diff. =20 Categ. 45 (2004), 193-240. Other papers on (weak) double categories, many of them by Bob Pare et =20= al., are referred to in the articles above. In book form Tom Leinster, Higher operads, higher categories, Cambridge Un. =20 Press 2004, has Section 5.2, devoted to weak double categories. Best regards Marco Marco Grandis Dipartimento di Matematica Universit=E0 di Genova Via Dodecaneso, 35 16146 Genova Italy e-mail: grandis@dima.unige.it tel: +39 010 353 6805 http://www.dima.unige.it/~grandis/ On 26 Oct 2005, at 22:08, John Baez wrote: > 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 > > > > > > --Apple-Mail-6--279414351 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1
Dear John,

The definition=A0of = weak=A0(or pseudo)=A0double category is what you would expect, once you = know the strict case and the definition of bicategory; think of a = "pseudocategory object in Cat". It has probably been in the categorical = folklore since quite a while.

You can find it=A0(including = the machinery of lax double functors, horizontal and vertical = transformations, etc.)=A0in two papers in Cahiers, where part of the = general theory of weak double categories has been = developed

=A0 =A0M. Grandis - R. Pare, Limits in double = categories, Cah. Topol. Geom. Diff. Categ. 40 (1999), = 162-220,

=A0 =A0- -, = Adjoints for double categories,=A0 Cah. Topol. Geom. Diff. Categ. = 45 (2004), 193-240.

Other papers = on (weak) double categories, many of them by Bob Pare et al., are = referred to in the articles above.
In book = form

=A0 =A0Tom = Leinster, Higher operads, higher categories, Cambridge Un. Press = 2004,

has=A0Section = 5.2, devoted to=A0weak double categories.

Best = regards


Marco = Grandis
Dipartimento di Matematica
Universit=E0 di = Genova
Via Dodecaneso, 35
16146 = Genova
Italy

e-mail: grandis@dima.unige.it
= tel: +39 010 353 6805
http://www.dima.unige.it/~gran= dis/

On 26 Oct 2005, at 22:08, John Baez = wrote:

Dear Categorists -

If you = weaken the notion of 2-category you get the notion of
bicategory.=A0 = 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?=A0 Did anyone
succeed?

Best,
jb






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