From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3451 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Reflexive coequalizers Date: Mon, 9 Oct 2006 21:09:46 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019312 8808 80.91.229.2 (29 Apr 2009 15:35:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:35:12 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Oct 9 21:30:11 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 09 Oct 2006 21:30:11 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GX5VP-0005iZ-IV for categories-list@mta.ca; Mon, 09 Oct 2006 21:29:59 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 12 Original-Lines: 70 Xref: news.gmane.org gmane.science.mathematics.categories:3451 Archived-At: Ah yes, I see now. I had observed that the underlying diagram of indiscrete graphs was still a coequalizer -- which unfortunately is to truncate one's simplicial sets a little too much! Many thanks for the enlightenment. Richard --On 09 October 2006 20:40 George Janelidze wrote: > Dear Richard, > > If we were asking the same question about the category SimplSet of > simplicial sets instead of Cat, the answer would be obvious since: > > (*) The functor Set ---> Set sending X to X^n preserves reflexive > coequalizers for each natural n. This (very simple) fact should be > considered as well known since it is involved in one of several well-known > proofs of monadicity of varieties of universal algebras over Set. > > (**) Since SimplSet is a Set-valued functor category, all colimits in it > reduce to colimits in Set. > > For those who are familiar with the standard adjunction between SimplSet and > Cat, the result for SimplSet will immediately imply the result for Cat > (since the fundamental category functor being the left adjoint in that > adjunction preserves all colimits and obviously sends "indiscrete simplicial > sets" to indiscrete categories). > > One could also use various (not too much) truncated simplicial sets instead > of the simplicial sets of course (e.g. what I once called "precategories"). > > I mean, I do not remember any reference, but I would not need the explicit > description of colimits in Cat. > > George Janelidze > > ----- Original Message ----- > From: "Richard Garner" > To: > Sent: Monday, October 09, 2006 12:14 PM > Subject: categories: Reflexive coequalizers > > >> >> Dear categorists, >> >> I have a proof that the indiscrete category functor >> Set -> Cat preserves reflexive coequalizers which, >> although straightfoward, uses the explicit >> description of colimits in Cat. Is this necessary, >> or can I deduce the result from general >> principles? >> >> [Also, I'm sure this result must appear somewhere >> but I can't find a reference for it. If anyone >> knows of one, I'd be grateful.] >> >> Many thanks, >> >> Richard >> >> >> > >