From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/891 Path: news.gmane.org!not-for-mail From: Philippe Gaucher Newsgroups: gmane.science.mathematics.categories Subject: Re: cogenerator in omegaCat ? Date: Wed, 21 Oct 1998 18:45:26 +0200 Message-ID: <199810211645.AA06093@irmast1.u-strasbg.fr> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241017296 28140 80.91.229.2 (29 Apr 2009 15:01:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:01:36 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Wed Oct 21 15:36:12 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA32690 for categories-list; Wed, 21 Oct 1998 14:30:58 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:891 Archived-At: > 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.