From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6346 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Communes paper, schismatic objects Date: Mon, 01 Nov 2010 10:44:23 -0700 Message-ID: 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: quoted-printable X-Trace: dough.gmane.org 1288649214 5673 80.91.229.12 (1 Nov 2010 22:06:54 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 1 Nov 2010 22:06:54 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Mon Nov 01 23:06:50 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 1PD2Wf-0000Cf-R3 for gsmc-categories@m.gmane.org; Mon, 01 Nov 2010 23:06:50 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58884) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PD2Vp-0005VF-67; Mon, 01 Nov 2010 19:05:57 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PD2Vk-00015v-Eh for categories-list@mlist.mta.ca; Mon, 01 Nov 2010 19:05:52 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6346 Archived-At: A couple of things. First, I neglected to mention that "Communes via=20 Yoneda, from an Elementary Perspective," Fundamenta Informaticae 123=20 (2010) 1=9616, DOI 10.3233/FI-2010-315 is about to appear and won't be=20 findable by Google just yet. Those interested in seeing it sooner can=20 find it on my site at http://boole.stanford.edu/pub/CommunesFundInf2010.pdf Second, as I said I wasn't passing judgment on the wisdom of avoiding=20 the term "schzophrenic" but merely pointing out the associated cost,=20 which needs to be balanced against the harm of any given word. So I followed Tom's pointer http://ncatlab.org/nlab/show/dualizing+object linking to a discussion of alternatives, which seemed inconclusive. Sam=20 (Staton?) made the point however that even if schizophrenia is not the=20 appropriate word, schizo is the appropriate prefix, having derived from=20 the Greek "split." So it is the medical condition that is inappropriately named, namely as=20 "split madness," with phrenitis and frenzy having a common origin. With that in mind it occurred to me that "schismatic" might be a=20 suitable alternative, as providing better continuity with the older=20 terminology by coming from the same root schizo, but more honestly so=20 than schizophrenia since in this case there really is a multiple=20 personality, and moreover there's nothing insane about it. (And it's a=20 syllable shorter to boot.) Third, while it is true that the schismatic object (to give the term a=20 trial run) is usually observed manifesting its split personality in=20 different categories, this is not the case in *-autonomous categories=20 where I and _|_ are the Jekyll and Hyde of the same category. (I=20 apologize to readers of this list with either of those surnames.) In all the examples I'm aware of, the two categories in which the=20 schismatic object occurs (once in each) admit a common completion to a=20 *-autonomous category which embeds one object as I and the other as _|_.=20 Considering them to be the "same" object found in two categories=20 misses the contravariance between them, which is brought out more=20 clearly by this joint completion, where they are clearly not the same=20 object but a pair of dual objects. My paper accounts for C.I. Lewis's qualia by viewing them as morphisms=20 running from I to _|_. If I and _|_ are rigid (|C(x,x)|=3D1) as for=20 Chu(Set,K)), the presence of a morphism from _|_ to I is inconsistent in=20 the sense that it collapses Hom(I,_|_) to a singleton, since I and _|_=20 respectively generate and cogenerate. So in order to have more than one=20 quale (in the Chu setting) there cannot be any morphism from _|_ to I.=20 (That was mainly in the nature of background on the neighborhood of I=20 and _|_.) Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]