From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5298 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: question on pseudomorphisms Date: Tue, 24 Nov 2009 12:13:29 +1100 Message-ID: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="ISO-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1259026672 29845 80.91.229.12 (24 Nov 2009 01:37:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 24 Nov 2009 01:37:52 +0000 (UTC) To: , categories Original-X-From: categories@mta.ca Tue Nov 24 02:37:45 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 1NCkLg-0005DQ-Nm for gsmc-categories@m.gmane.org; Tue, 24 Nov 2009 02:37:44 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NCk5a-0000Vr-4V for categories-list@mta.ca; Mon, 23 Nov 2009 21:21:06 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5298 Archived-At: Dear Albert, In the case where T is a 2-monad with rank, and the pseudomorphisms are (as usual) assumed to be coherent, then this was proved by Blackwell-Kelly-Powe= r 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 the 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. On 23/11/09 11:55 PM, "burroni@math.jussieu.fr" wrote: > Dear all, >=20 > Has the following question been already studied and, if it is the case > (it is my opinion), where ? > Let T be a monad on Cat (not necessarily a 2-monad), C the category of > T-algebras and C' the cat=E9gorie with the same objets but with > pseudomorphisms (morphisms up to natural isomorphisms --- eventually > with coherences). > The inclusion i : C --> C' has a left adjoint j : C' --> C. >=20 > The question is : for all T-algebra A, is the canonical morphisms m : > A --> i(j(A)) an equivalence (of the underlying categories) ? >=20 > Regards, > Albert >=20 >=20 >=20 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]