From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/894 Path: news.gmane.org!not-for-mail From: carlos@picard.ups-tlse.fr (Carlos Simpson) Newsgroups: gmane.science.mathematics.categories Subject: Re: cogenerator in omegaCat ? Date: Wed, 21 Oct 1998 22:47:13 GMT Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241017300 28159 80.91.229.2 (29 Apr 2009 15:01:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:01:40 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Thu Oct 22 16:46:48 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA17573 for categories-list; Thu, 22 Oct 1998 15:42:39 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Eudora F1.5.4 Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 48 Xref: news.gmane.org gmane.science.mathematics.categories:894 Archived-At: In response to Ph. Gaucher's question: I (try to, at least...) treat this question for weak $n$-categories in my preprint ``Limits in $n$-categories'', available on the xxx preprint server as alg-geom 9708010. If I understand correctly, the set-theoretical problem you raise is the same as the one encountered in section 5 of my preprint. The conclusion is that the (weak) $n+1$-category $nCAT$ is closed under direct limits. It seems that coproducts of strict $n$-categories, if they exist, cannot actually be the ``right'' ones because in that case, every weak $n$-category would be equivalent to a strict one. I haven't made this argument rigorous, though. ---Carlos Simpson PS what is a ``comma category'' or ``comma object''? > >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.