From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7750 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: Isbell & MacLane on the insufficiency on skeletal categories Date: Fri, 7 Jun 2013 01:00:23 -0400 Message-ID: References: Reply-To: Colin McLarty NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1370720862 5596 80.91.229.3 (8 Jun 2013 19:47:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 8 Jun 2013 19:47:42 +0000 (UTC) Cc: "categories@mta.ca" To: Vaughan Pratt Original-X-From: majordomo@mlist.mta.ca Sat Jun 08 21:47:43 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 1UlP70-00023j-Ql for gsmc-categories@m.gmane.org; Sat, 08 Jun 2013 21:47:42 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38824) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UlP5m-00031x-EE; Sat, 08 Jun 2013 16:46:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UlP5m-0002nZ-Jh for categories-list@mlist.mta.ca; Sat, 08 Jun 2013 16:46:26 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7750 Archived-At: Vaughan, Your quote runs two of my paragraphs together. .Certainly the point about N does not need Isbell's argument. Colin On Thu, Jun 6, 2013 at 6:00 PM, Vaughan Pratt wrote: > 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/ ]