From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5300 Path: news.gmane.org!not-for-mail From: burroni@math.jussieu.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: question on pseudomorphisms Date: Tue, 24 Nov 2009 20:50:13 +0100 Message-ID: References: Reply-To: burroni@math.jussieu.fr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1259108520 10093 80.91.229.12 (25 Nov 2009 00:22:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 25 Nov 2009 00:22:00 +0000 (UTC) To: Steve Lack , categories Original-X-From: categories@mta.ca Wed Nov 25 01:21:53 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1ND5do-0008Ij-Ki for gsmc-categories@m.gmane.org; Wed, 25 Nov 2009 01:21:52 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ND53X-0005MA-Jq for categories-list@mta.ca; Tue, 24 Nov 2009 19:44:23 -0400 In-Reply-To: Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5300 Archived-At: Dear Steve, Thank you for your precious informations (I have never read the =20 article "two-dimensional ..."). But my interest is really in the case of non necessarily 2-monad over =20 Cat and I am convinced that the adjonction j -| i exists, with or =20 without coherences (for the pseudomorphisms). Best regards, Albert Steve Lack a =E9crit=A0: > Dear Albert, > > In the case where T is a 2-monad with rank, and the pseudomorphisms are (a= s > usual) assumed to be coherent, then this was proved by Blackwell-Kelly-Pow= er > in the paper 2-dimensional monad theory. (Here "rank" means that the > 2-functor T preserves alpha-filtered colimit for some alpha - without some > such assumption, I don't know how you can prove that the left adjoint j > exists, and I suspect it does not.) > > If T has rank but pseudomorphisms are not required to be coherent, then th= e > adjoint j will exist, but the morphism A-->ijA need not be an equivalence > (take the identity monad for example). > > If T is not even a 2-monad then I'm not sure what coherence of the 2-cells > would mean, but in any case there will be problems. > > Regards, > > Steve Lack. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]