From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4979 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Fundamental Theorem of Category Theory? Date: Wed, 17 Jun 2009 07:58:36 +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 1245246805 19428 80.91.229.12 (17 Jun 2009 13:53:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 17 Jun 2009 13:53:25 +0000 (UTC) To: Vaughan Pratt , categories Original-X-From: categories@mta.ca Wed Jun 17 15:53:23 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 1MGvZp-0000IU-HN for gsmc-categories@m.gmane.org; Wed, 17 Jun 2009 15:53:21 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MGuqq-0006e1-Ke for categories-list@mta.ca; Wed, 17 Jun 2009 10:06:52 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4979 Archived-At: Dear Vaughan, Your proposed characterization is actually a characterization of full subcategories of [J^op,Set] containing the representables. To get the whole presheaf category you should add that C is cocomplete, and that homming out of objects in J is cocontinuous (i.e. C(j,-) is cocontinuous for all j in J). Steve. On 16/06/09 7:58 AM, "Vaughan Pratt" wrote: > Apropos of the Yoneda Lemma, is there some reason why it is usually > stated on its own rather than as one direction of a characterization of > categories of presheaves on J? Unless I've overlooked or misunderstood > something it seems to me that the Yoneda Lemma should state that C is a > category of presheaves on J if and only if there exists a full, > faithful, and dense functor from J to C. > > This should generalize the characterization of an Archimedean field as > any dense extension of the rationals. > > Vaughan Pratt > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]