From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4241 Path: news.gmane.org!not-for-mail From: peasthope@shaw.ca Newsgroups: gmane.science.mathematics.categories Subject: Re: A small cartesian closed concrete category Date: Mon, 03 Mar 2008 10:37:02 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019816 12366 80.91.229.2 (29 Apr 2009 15:43:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:43:36 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Mar 3 10:42:32 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 03 Mar 2008 10:42:32 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWBmo-0005g9-5m for categories-list@mta.ca; Mon, 03 Mar 2008 10:37:02 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 77 Xref: news.gmane.org gmane.science.mathematics.categories:4241 Archived-At: Folk, At Thu, 14 Feb 2008 15:06:49 -0500 I wrote, "Is there a cartesian closed concrete category which=20 is small enough to write out explicitly?" =20 At Fri, 15 Feb 2008 08:47:57 +0000 Philip Wadler srote, "... please summarize the replies ... and send ... to the ... list? ... interested to see if you receive a positive reply." I've counted 16 respondents! The question is=20 answered well. With my limited knowledge, the=20 summary probably fails to credit some of the=20 responses adequately but this is not intentional. Thanks to everyone who replied! 5 messages mentioned Hyting-algebras. Never heard of them. Lawvere & Schanuel=20 do not mention them in the 1997 book. =20 Will store the terms for future reference. Fred Linton wrote, "... skeletal version of the full category ... having as only objects the ordinal numbers 0 and 1. Here 0 x A =3D 0, 1 x A =3D A, 0^1 =3D 0, 0^0 =3D 1, 1^A =3D 1. In other words, B x A =3D min(A, B), B^A =3D max(1-A, B)." My product diagrams are at=20 http://carnot.yi.org/category01.jpg . Now I can try to illustrate the uniqueness=20 of map objects according to L&S, page page 314,=20 Exercise 1. Does this category have a name? =20 Is Boolean Category sensible? Two messages mentioned lambda calculus. Another topic for future reference. Stephen Lack asked "How small is small?=20 How explicit is explicit?" Probably=20 several other readers wondered the same. Fred's reply is small enough and explicit=20 enough to write out in detail. One message addressed the term "concrete". =20 I referred to Concrete Categories in the=20 Wikipedia. Matt Hellige mentioned categories a little=20 bigger than that described by Fred. =20 For instance, objects 0, 1, 2, 3. Map A -> B exists iff A < B. B x A =3D? min(A, B) =20 I should sketch the details of some of these=20 examples beyond the 0, 1 case above. Andrej Bauer described Fred's category in the context=20 of Heyting algebra and noted a proof by=20 Peter Freyd. Thorsten Altenkirch mentioned an equational=20 inconsistency which is beyond my present=20 grasp. Apologies to anyone who's reply is not =20 addressed adequately. If someone requests,=20 I can revise the summary and resubmit it. Thanks, ... Peter E. Desktops.OpenDoc http://carnot.yi.org/