From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6360 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Communes paper, schismatic objects Date: Wed, 03 Nov 2010 15:35:17 -0700 Organization: Stanford University Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1288924113 19018 80.91.229.12 (5 Nov 2010 02:28:33 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 5 Nov 2010 02:28:33 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Fri Nov 05 03:28:18 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PEC2L-0007nN-R6 for gsmc-categories@m.gmane.org; Fri, 05 Nov 2010 03:28:18 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:49894) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PEC1n-0001EO-RF; Thu, 04 Nov 2010 23:27:43 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PEC1l-0002Qp-8U for categories-list@mlist.mta.ca; Thu, 04 Nov 2010 23:27:41 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6360 Archived-At: On 11/1/2010 4:52 PM, Todd Trimble wrote: > It is both at once: a Boolean algebra > object in the category of compact Hausdorff spaces, and we need both > forms in the same body so that we can say hom_{CH}(-, 2) is a Boolean > algebra valued functor. Your example perfectly illustrates my point about I and _|_ being distinct but dual objects. In CH, I = 1 and _|_ = 1+1 (I'm assuming by 2 you mean 1+1 rather than the Sierpinski space). The contravariant Boolean algebra valued functor hom_{CH}(-, 1+1): CH^op --> Bool sends I and _|_ in CH to respectively _|_ and I in Bool. In both categories I is the free object on one generator and as such a generator and the tensor unit, while its dual _|_ is cofree, a cogenerator, and the unit for par (to the extent tensor and par are defined in each category -- they become fully defined in a common self-dual unification that covariantly embeds both categories, namely Chu(Set,2)). Understood via the above functor as Boolean algebra objects, in CH I = 1 and _|_ = 1+1 are respectively the 2-element and 4-element Boolean algebras, while in Bool these are interchanged: I has 4 elements (the free Boolean algebra on one generator) and _|_ has 2. In both categories _|_ is the dualizing object. I would not say that the 2-element and 4-element Boolean algebras are the same. In my book they are distinct. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]