From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5933 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: pullback of locally presentable categories Date: Thu, 17 Jun 2010 15:38:25 +1000 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1276859359 2036 80.91.229.12 (18 Jun 2010 11:09:19 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 18 Jun 2010 11:09:19 +0000 (UTC) Cc: Gaucher Philippe , categories To: Richard Garner Original-X-From: categories@mta.ca Fri Jun 18 13:09:15 2010 connect(): No such file or directory 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 4.69) (envelope-from ) id 1OPZRi-0001xT-4i for gsmc-categories@m.gmane.org; Fri, 18 Jun 2010 13:09:14 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OPYwp-0001nW-OZ for categories-list@mta.ca; Fri, 18 Jun 2010 07:37:19 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5933 Archived-At: Dear Richard and others, On 17/06/10 12:35 AM, "Richard Garner" wrote: > > ---and I have further noticed that, although the first > situation may encompass the other two, this is by the by, > since the argument in this first situation is incorrect. I > have only shown the vertex of the pullback to be accessible, > but not necessarily locally presentable. I don't see any > obvious way of rectifying this. However, the argument in the > second and third situations is still valid. So to > summarise:-- > > Given a cospan of locally presentable categories and > accessible functors, > - if one of the functors is an isofibration, then the > pullback is accessible; Yes, I agree. > - if one of the functors is monadic or comonadic, then the > pullback is locally presentable. No, I don't think this is enough for local presentability. If (as well as both functors being accessible and one being an isofibration) both functors were continuous then the pullback would be complete and so locally presentable. If both were cocontinuous, then the pullback would be cocomplete and so locally presentable. But monadicity of one is not enough. For example, let A be the full subcategory of Set consisting of all one-element sets. The inclusion and both A and Set are locally presentable. Now choose any functor 1-->Set whose image is not a singleton. This is accessible, and 1 is locally presentable. The pullback of A->Set and 1->Set is empty, so not locally presentable. Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]