From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5023 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: no fundamental theorems please Date: Wed, 24 Jun 2009 16:17:02 +0100 Message-ID: Reply-To: Steve Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1246005023 4962 80.91.229.12 (26 Jun 2009 08:30:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 26 Jun 2009 08:30:23 +0000 (UTC) To: Vaughan Pratt , Categories Original-X-From: categories@mta.ca Fri Jun 26 10:30:16 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 1MK6p5-0006C8-PJ for gsmc-categories@m.gmane.org; Fri, 26 Jun 2009 10:30:15 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MK61c-0005vU-6I for categories-list@mta.ca; Fri, 26 Jun 2009 04:39:08 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5023 Archived-At: Vaughan Pratt wrote: > ... The Yoneda Lemma usually states that J embeds (meaning > fully) in [J^op,Set], ... Dear Vaughan, The usual statement is significantly stronger than that (see e.g. Mac Lane, Mac Lane and Moerdijk, or Wikipedia). It says that, for contravariant functors F: C -> Set, the elements of FX are in bijection with transformations to F from the representable functor for X. Your statement can be deduced by considering the particular case where F too is representable. (To put it another way, the representable presheaf for X is freely generated - as presheaf - by a single element (the identity morphism) at X. This then allows you to calculate the left adjoint of the forgetful functor from presheaves over C to ob(C)-indexed families of sets.) There can be no doubt that this strong Yoneda Lemma is vitally important when calculating with presheaves - for example, it shows immediately how to calculate exponentials and powerobjects. If F and G are two presheaves, then the exponential G^F is calculated by G^F(X) = nt(Y(X), G^F) (by Yoneda's Lemma) = nt(Y(X) x F, G) (by definition of exponential) I don't think you can get it and its useful consequences from your weaker statement, even if you start strengthening yours in the way you suggest by supplying converses. Another closely related and important result, though not known as Yoneda's Lemma as far as I know, is that the presheaf category over C is a free cocompletion of C, and the Yoneda embedding is the injection of generators. (By the way, I agree that category theory doesn't have to have a Fundamental Theorem. I haven't see any compelling reason to appoint one.) Regards, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]