From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7747 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Isbell & MacLane on the insufficiency on skeletal categories Date: Thu, 06 Jun 2013 15:00:38 -0700 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: plane.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 1370562206 23153 80.91.229.3 (6 Jun 2013 23:43:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 6 Jun 2013 23:43:26 +0000 (UTC) Cc: "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Fri Jun 07 01:43:27 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 1Ukjpz-0003Fd-Lu for gsmc-categories@m.gmane.org; Fri, 07 Jun 2013 01:43:23 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37749) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Ukjon-0003oR-MQ; Thu, 06 Jun 2013 20:42:09 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ukjoo-0003Vg-Ir for categories-list@mlist.mta.ca; Thu, 06 Jun 2013 20:42:10 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7747 Archived-At: On 5/25/2013 8:47 AM, Colin McLarty wrote: > Of course is also follows that NxN=N. But it does not follow, and in fact > it is refutable, that the projection functions are the identity function > 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. The inconsistency found by Isbell works even when the projections seem quite reasonable, namely when taken to be the three projections from the ternary product XxYxZ whose elements are triples (x,y,z) (as the necessary meaning of identity of associativity a: Xx(YxZ) --> (XxY)xZ). One can in fact consistently equip Skel(FinSet) with such structure. Isbell shows that extending this to infinite sets breaks down, namely by creating additional equations not encountered with finite sets due to interference between binary and ternary product resulting from the identification of N with NxN. One can get close to a skeleton of Set using tau-categories as per section 1.493 of Cats & Alligators; N becomes the ordinal omega, whose square is a distinct object albeit still isomorphic to omega. The full subcategory of finite ordinals is (isomorphic to) Skel(FinSet). Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]