From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4990 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Fundamental Theorem of Category Theory? Date: Thu, 18 Jun 2009 01:33:58 -0700 Message-ID: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1245320976 10606 80.91.229.12 (18 Jun 2009 10:29:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 18 Jun 2009 10:29:36 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Thu Jun 18 12:29:32 2009 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 4.50) id 1MHEs5-0002mZ-Gq for gsmc-categories@m.gmane.org; Thu, 18 Jun 2009 12:29:29 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MHE8m-00058A-K3 for categories-list@mta.ca; Thu, 18 Jun 2009 06:42:40 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4990 Archived-At: On 6/17/2009 3:45 PM, Steve Lack wrote: > Hmm. Not sure if you mean you're allowing any full subcategory of > [J^op,Set]; if so then you should drop the requirement that J-->C be fully > faithful. By "category of presheaves on J" I had in mind retaining J as part of it. >> Am I missing something? I was thinking that followed from density of J >> in C. >> > > No. The category Setf of finite sets has a fully faithful dense inclusion in > to the (presheaf) category Set of all sets, but Set is not [Setf^op,Set]. Oops, right, I was mixing up cocomplete and cocompletion-of. (Actually I don't think in terms of either, I find it easier to think of [J^op,Set] as the maximal dense extension of J up to equivalence, in the sense that all dense extensions of J are full subcategories of it.) Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]