From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7097 Path: news.gmane.org!not-for-mail From: Jean Benabou Newsgroups: gmane.science.mathematics.categories Subject: Re: Dualities arising via pairs of schizophrenic objects Date: Sun, 4 Dec 2011 15:25:29 +0100 Message-ID: References: Reply-To: Jean Benabou NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1323094918 32345 80.91.229.12 (5 Dec 2011 14:21:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 5 Dec 2011 14:21:58 +0000 (UTC) To: Fred Linton , Categories Original-X-From: majordomo@mlist.mta.ca Mon Dec 05 15:21:52 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 1RXZQV-000666-MQ for gsmc-categories@m.gmane.org; Mon, 05 Dec 2011 15:21:51 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46055) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RXZOY-0000aF-EY; Mon, 05 Dec 2011 10:19:50 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RXZOW-00086o-SH for categories-list@mlist.mta.ca; Mon, 05 Dec 2011 10:19:48 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7097 Archived-At: Dear Fred, I keep your notations. The concreteness of A is far from enough to justify the definition you give, namely: > 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. Eckman and Hilton considered only the case when A is a category of essentially algebraic (i.e. definable by projective limits) structures over Sets. In more general cases, it just doesn't work. You can't even prove, for X=A, that an object G of X bears the structure of an object of A. Take for A the concrete category of totally ordered sets and order- preserving functions, with U the obvious forgetful functor. The same is true for A=Fields, A=Topological Spaces, A=Finite Sets, etc., to take very simple examples. Bien amicalement, Jean > > 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/ ]