From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/892 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: cogenerator in omegaCat ? Date: Wed, 21 Oct 1998 15:36:24 -0400 (EDT) Message-ID: References: <199810211645.AA06093@irmast1.u-strasbg.fr> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017298 28147 80.91.229.2 (29 Apr 2009 15:01:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:01:38 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Thu Oct 22 11:22:19 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA24395 for categories-list; Thu, 22 Oct 1998 08:54:31 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: <199810211645.AA06093@irmast1.u-strasbg.fr> Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 46 Xref: news.gmane.org gmane.science.mathematics.categories:892 Archived-At: I imagine that omega-categories, however defined, will be a locally c-presentable category (c=cardinal of the continuum) in the sense of Gabriel-Ulmer, equivalently complete and c-accessible in the sense of Makkai-Pare and hence cocomplete. In other words, the colimit will grow but not by much. Actually, aleph_1 is all you are really going to need. On Wed, 21 Oct 1998, Philippe Gaucher wrote: > > > > No, it seems not since a co-generator for omega cat > > would surely give rise to one for cat in particular, but such > > does not exist. This contrasts with the situation for the > > "larger" universe of simplicial sets. A category of "small" > > sets is a kind of approximation to a co-generator, but each > > enlargement of the meaning of "small" creates new categories > > which are not co-generated. > > > The argument sounds reasonable. Before this question, I was > convinced of the existence of this cogenerator. I have to find > something else for the lemma I would like to prove... > > Since it does not exist, I have another questions (I suppose well- > known) and any reference abou the subject would be welcome : > > How does one prove the cocompleteness of omegaCat (small & strict) ? > The only idea of proof I had in mind until this question was : omegaCat > is obviously complete (and the forgetful functor towards the category of Sets > preserves projective limits), and well-powered and a cogenerator > => the cocompleteness (Borceux I, prop 3.3.8 p 112). > > Without cogenerator, how can one prove the cocompleteness ? The explicit > construction of the colimit seems to be very hard : the forgetful > functor towards Set does not preserve colimits because the > underlying set of the colimit might be bigger than the colimit of the > underlying sets. Every time two n-morphisms are identified in the > colimit of the underlying sets, p-morphisms (with p>n) might be "created" > by the colimit. > > Thanks in advance for any answer. pg. > >