From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4215 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Re: A small cartesian closed concrete category Date: Sat, 16 Feb 2008 01:17:24 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019798 12248 80.91.229.2 (29 Apr 2009 15:43:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:43:18 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Feb 15 21:24:51 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 15 Feb 2008 21:24:51 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JQBer-00034b-Ja for categories-list@mta.ca; Fri, 15 Feb 2008 21:16:01 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 37 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:4215 Archived-At: PETER EASTHOPE wrote: > Is there a cartesian closed concrete category which > is small enough to write out explicitly? It would be > helpful in learning about map objects, exponentiation, > distributivity and etc. Can such a category be made > with binary numbers for instance? A Heyting algebra, viewed as a category (every poset is a category), is a CCC. If you take a small Heyting algebra, e.g. the topology of a finite topological space, you can write it out explicitly. If you would like a CCC made from n-bit binary numbers, here is how you can do it: The two-point lattice B = {0, 1} is a Boolean algebra with the usual logical connectives as the operations. Because B is a poset with 0<=1, it is also a category (with two objects 0, 1 and a morphism between them). Since every Boolean algebra is a Heyting algebra, B is cartesian closed, with the following structure: - 1 is the terminal object - the product X x Y is the conjuction X & Y - the exponential Y^X is the implicatoin X => Y The product of n copies of B is the same thing as n-tuples of bits, i.e., the n-bit numbers. This is again a CCC (with coordinate-wise structure). At this late hour I cannot see what can be said about finite CCC's which are not (eqivalent to) posets. Andrej