From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4989 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Fundamental Theorem of Category Theory? Date: Thu, 18 Jun 2009 08:45:25 +1000 Message-ID: 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: ger.gmane.org 1245282961 18399 80.91.229.12 (17 Jun 2009 23:56:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 17 Jun 2009 23:56:01 +0000 (UTC) To: Vaughan Pratt , categories Original-X-From: categories@mta.ca Thu Jun 18 01:55:59 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 1MH4z0-0007pY-Sj for gsmc-categories@m.gmane.org; Thu, 18 Jun 2009 01:55:59 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MH4Po-00062c-Fo for categories-list@mta.ca; Wed, 17 Jun 2009 20:19:36 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4989 Archived-At: On 17/06/09 1:28 PM, "Vaughan Pratt" wrote: > Steve Lack wrote: >> Your proposed characterization is actually a characterization of full >> subcategories of [J^op,Set] containing the representables. > > Right, that's what I meant by "*a* category of presheaves on J" (as > opposed to *the* category of all presheaves on J), the point of my > analogy with Archimedean fields (as opposed to the field of all reals). > 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. >> To get the whole >> presheaf category you should add that C is cocomplete, > > Right, just as to get all of the reals one should say that the > Archimedean field is complete. For situations where one doesn't need > the whole thing it is convenient to be able to characterize the > categorical counterpart of an Archimedean field, with J in place of Q, > as any full, faithful and dense extension of J. Density serves to keep > the extension inside [J^op,Set], just as it keeps Archimedean fields > inside R. > >> and that homming out >> of objects in J is cocontinuous (i.e. C(j,-) is cocontinuous for all j in >> J). > > 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]. Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]