From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2174 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Category of Heyting Algebras Date: Wed, 12 Feb 2003 11:31:57 -0800 Message-ID: <20030212193157.GB19717@math-rs-n01.ucr.edu> References: <20030211214817.55877.qmail@web12203.mail.yahoo.com> <200302121220.09792.mcgrail@bard.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018466 2862 80.91.229.2 (29 Apr 2009 15:21:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:21:06 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Feb 13 14:52:31 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Feb 2003 14:52:31 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18jONv-0006WD-00 for categories-list@mta.ca; Thu, 13 Feb 2003 14:46:59 -0400 Content-Disposition: inline In-Reply-To: <200302121220.09792.mcgrail@bard.edu> User-Agent: Mutt/1.4i Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 29 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:2174 Archived-At: Robert McGrail wrote: >I assume that in a Heyting algebra T does not equal F. >This follows the intuitive introduction of Heyting algebras by >Moerdijk/MacLane as capturing the algebraic structure of topologies. Probably there are people that put this in the definition -- the same people that require 0 != 1 in any ring, or that a topological space not be empty, or (for that matter) that a CCC not be trivial. But if you're not one of those people, then the Heyting algebra {*} captures the algebraic structure of the (unique topology on the) empty space. >Anyway, under these assumptions, the trivial HA {T,F} is both initial and >final. So with my definition, {T,F} is initial but {*} is final. But even with yours, I don't believe that {T,F} becomes final. There are 2 homomorphisms to it from the power set of {T,F} (as a Boolean algebra, which is a special kind of Heyting algebra). I don't think that your category has a final object (any more than the category of nontrivial rings does, nor the category of nonempty spaces has an initial object). -- Toby