From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5021 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: no fundamental theorems please Date: Tue, 23 Jun 2009 11:58:26 -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 1245789784 17175 80.91.229.12 (23 Jun 2009 20:43:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 23 Jun 2009 20:43:04 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Tue Jun 23 22:42:57 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 1MJCpU-0003aJ-2o for gsmc-categories@m.gmane.org; Tue, 23 Jun 2009 22:42:56 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MJCBZ-0003Bu-Fq for categories-list@mta.ca; Tue, 23 Jun 2009 17:01:41 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5021 Archived-At: On 6/22/2009 2:07 PM, Paul Taylor wrote: > Claudio, Vaughan, Ross and others have mentioned various more > "sophisticated" versions of the Yoneda Lemma. I don't see how supplying the missing half of an if-and-only-if is "more sophisticated." The Yoneda Lemma usually states that J embeds (meaning fully) in [J^op,Set], but it could just as well state this for any factor C between J and [J^op,Set]. In such situations it is natural to ask whether the converse holds; it doesn't, but if one adds to full-and-faithful the (very natural) requirement of density then it does: the dense extensions of J are precisely the categories of presheaves on J, by which I mean the full subcategories of [J^op,Set] that retain J as a full subcategory (the sense of "on"). I don't call that sophisticated. > However, I contend that the more sophisticated the result is, > the LESS it deserves to be called "the fundamental theorem of > category theory". Indeed. Any branch of mathematics that does so only contributes to the image of mathematics as a difficult subject. > I particularly like Euclid's algorithm for the highest common > factor as a historical and methodological example, Without > meaning to dictate to number theorists how their subject should > be organised, let me suggest for the sake of argument that it > is a pretty good candidate for being called the "fundamental > theorem of number theory". That's an algorithm. The relevant theorem is also called the fundamental theorem of arithmetic. One could state the essential idea in sophisticated language as "Z is a principal ideal domain" but the more usual statement about uniqueness of factorization of positive integers makes number theory a more accessible subject. > As I say, I like Euclid's algorithm because the idea has > survived many many revolutions in the "official" foundations > of mathematics. > [...] Cantor's "theorem" about a powerset being > strictly bigger than a set. This belongs entirely to the dogma > of set theory. When set theory is overturned, this miserable > and wholely misguided "theorem" will go in the dustbin of > mathematical history with it. I like Cantor's theorem because, like Euclid's algorithm, it will survive the revolution Paul is trying to foment here. > But Euclid's algorithm will live forever. As will diagonalization arguments like Cantor's. > But [Yoneda's Lemma] still shouldn't be called the "fundamental theorem"! Unlike the Fundamental Theorems of Arithmetic and of Algebra, the Yoneda Lemma has not yet established itself as *the* fundamental theorem of category theory. Nor will it unless a reasonable consensus to that effect emerges. I suggest waiting a few years before coming to any conclusion about whether CT has an FT, and if so what it is. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]