From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7096 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Dualities arising via pairs of schizophrenic objects Date: Fri, 02 Dec 2011 10:59:12 -0500 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1322947877 12937 80.91.229.12 (3 Dec 2011 21:31:17 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 3 Dec 2011 21:31:17 +0000 (UTC) Cc: David Roberts To: Categories Original-X-From: majordomo@mlist.mta.ca Sat Dec 03 22:31:12 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RWxAu-0008Cx-JW for gsmc-categories@m.gmane.org; Sat, 03 Dec 2011 22:31:12 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:43211) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RWx6D-0003dD-Vb; Sat, 03 Dec 2011 17:26:22 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RWx6C-0000ty-3r for categories-list@mlist.mta.ca; Sat, 03 Dec 2011 17:26:20 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7096 Archived-At: On Fri, 02 Dec 2011 08:35:30 AM EST, David Roberts wrote: = > ... always thought it odd that even when one wants to accept > category-theoretic > foundations (e.g. ETCS or similar), then suddenly something like this > comes along, > where people start saying there is a thing which is an object of two > different categories. Ever since Eckmann-Hilton, and perhaps even before, the notion of an object G in one category X bearing the structure of an object in some concrete other category A (concrete via U: A -> Sets, say) has been = clearly and unambiguously expressed as follows: The hom functor X(-, G): X^op -> Sets is given a factorization thru' U. If both X and A are concrete, it's perfectly plausible for an object of X to bear the structure of an object in A, and vice versa, and a brief peek at the example of 2 as BA w/ KT_2-space structure and as KT_2-space with BA structure will make short work of understanding how an object may be thought of as "inhabiting both categories at once": indeed, it's that contravariant adjoint pair alone, between A and X, that provides the duality in John Isbell's 1972 approach, where = at most one of A and X need be concrete. HTH. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]