From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3017 Path: news.gmane.org!not-for-mail From: jean benabou Newsgroups: gmane.science.mathematics.categories Subject: Re: horizontal composition Date: Tue, 31 Jan 2006 20:22:16 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v543) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019047 6908 80.91.229.2 (29 Apr 2009 15:30:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:47 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Wed Feb 1 12:28:37 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Feb 2006 12:28:37 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1F4Kha-0001oY-80 for categories-list@mta.ca; Wed, 01 Feb 2006 12:19:26 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 43 Original-Lines: 62 Xref: news.gmane.org gmane.science.mathematics.categories:3017 Archived-At: I thought I had invented bicategories in 1967, and that, at the very=20= beginning of the paper, in =A71, I had defined the two composition laws=20= and drawn pictures to explain them. Of course I denoted by capital=20 letters the 1-cells, thinking of functors, and by small letters the=20 2-cells, thinking of natural transformations. That certainly makes a=20 tremendous difference with Susan Niefield's notation who uses the=20 converse convention and amply justifies Marco Grandis in giving=20 references dated 1994 and 1996, i.e. more than 25 years posterior to my=20= original paper. With best regards > > You can find the strict version of that result in Prop. 1.4 of > > - M. Grandis, Homotopical algebra in homotopical categories, Appl.=20 > Categ. > Structures 2 (1994), 351-406. > > I do not know if it has been written down elsewhere. > > For sure, whiskering of natural transformations with functors is used=20= > in: > > - R. Street, Categorical structures, in: Handbook of Algebra, Vol. 1,=20= > 529-577, > North Holland, Amsterdam 1996. > > where you can find the notion of a sesqui-category (which does not=20 > assume the > "reduced interchange axiom" you are mentioning). > > With best regards > > M. Grandis > >> >> Does anyone know of a reference for the following definition of a >> bicategory? The primitive composites are: >> >> gf for composable 1-cells >> GF for vertically composable 2-cells >> f*G and F*g for horizontally composable pairs of each >> >> with appropriate axioms including (G*f')(g*F)=3D(g'*F)(G*f), for >> F:f->f':X->Y and G:g->g':Y->Z. The horizontal composite G*F is=20 >> defined to >> be the common value of the two vertical composites. >> >> -Susan > > >