From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7749 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Isbell & MacLane on the insufficiency on skeletal categories Date: Thu, 06 Jun 2013 23:15:08 -0400 Message-ID: <909RFgDoi6000S04.1370574908@web04.cms.usa.net> Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1370720787 4805 80.91.229.3 (8 Jun 2013 19:46:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 8 Jun 2013 19:46:27 +0000 (UTC) Cc: "categories@mta.ca" To: Vaughan Pratt Original-X-From: majordomo@mlist.mta.ca Sat Jun 08 21:46:28 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UlP5l-0001Ae-30 for gsmc-categories@m.gmane.org; Sat, 08 Jun 2013 21:46:25 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38815) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UlP4X-0002sJ-Fp; Sat, 08 Jun 2013 16:45:09 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UlP4X-0002lj-LB for categories-list@mlist.mta.ca; Sat, 08 Jun 2013 16:45:09 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7749 Archived-At: On Thu, 06 Jun 2013 07:45:03 PM EDT, by Vaughan Pratt : > On 5/25/2013 8:47 AM, Colin McLarty wrote: >> Of course is also follows that NxN=3DN. But it does not follow, and = in fact >> it is refutable, that the projection functions are the identity funct= ion >> 1_N. Isbell's argument is on p. 164 of my copy of CfWM (1998). > = > Why do you need Isbell's long argument, or even any monoidal structure > on Set, to obtain a contradiction here? Just use that NxN is a product= > and observe that the pair (3,4) in NxN (as a map from 1 to NxN) would > have to be both 3 and 4 (as maps from 1 to N) when the projections are > the identity. Even more convincing: The equalizer of those two projections from N x N must be the diagonal in N x N. But if those projections are equal, their equalizer is all of N x N. Thus every map to N x N factors through the diagonal there, i.e., no matter what the object A, for every pair of maps= f, g: A --> N, we must have f =3D g. It will follow that N is terminal. [Or was that your argument, Vaughan, that I somehow did not recognize?] = Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]