From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4213 Path: news.gmane.org!not-for-mail From: "Matt Hellige" Newsgroups: gmane.science.mathematics.categories Subject: Re: A small cartesian closed concrete category Date: Fri, 15 Feb 2008 13:08:11 -0600 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019797 12237 80.91.229.2 (29 Apr 2009 15:43:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:43:17 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Feb 15 15:23:16 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 15 Feb 2008 15:23:16 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JQ64W-0003yi-Ed for categories-list@mta.ca; Fri, 15 Feb 2008 15:18:08 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 35 Original-Lines: 27 Xref: news.gmane.org gmane.science.mathematics.categories:4213 Archived-At: On Thu, Feb 14, 2008 at 9:46 PM, Fred E.J. Linton wrote: > On Thu, 14 Feb 2008 10:07:27 PM EST, PETER EASTHOPE > asked: > > > Is there a cartesian closed concrete category which > > is small enough to write out explicitly? > > try the skeletal version of the full category of "sets of cardinality < 2" > having as only objects the ordinal numbers 0 and 1. > > Here 0 x A = 0, 1 x A = A, 0^1 = 0, 0^0 = 1, 1^A = 1. > In other words, B x A = min(A, B), B^A = max(1-A, B). > Or, in case that's too small, what about any short chain? For instance, let S = {0,1,2,3} and say there exists a morphism a -> b iff a < b. I believe this is cartesian closed, and I believe it can easily be understood as concrete. This should be enough to give non-trivial product and exponentiation, but you can still draw the whole diagram. Matt -- Matt Hellige / matt@immute.net http://matt.immute.net