From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7028 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: The boringness of the dual of exponential Date: Sun, 6 Nov 2011 22:55:08 +0100 Message-ID: References: Reply-To: Thomas Streicher NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1320620693 26321 80.91.229.12 (6 Nov 2011 23:04:53 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 6 Nov 2011 23:04:53 +0000 (UTC) Cc: categories To: David Leduc Original-X-From: majordomo@mlist.mta.ca Mon Nov 07 00:04:47 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 1RNBlf-0008Hx-7W for gsmc-categories@m.gmane.org; Mon, 07 Nov 2011 00:04:47 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42334) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RNBk6-00004s-LR; Sun, 06 Nov 2011 19:03:10 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RNBk4-0007xD-TH for categories-list@mlist.mta.ca; Sun, 06 Nov 2011 19:03:08 -0400 Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7028 Archived-At: > I have a conjecture: the dual of exponential is boring in any > category. Is there a counterexample to this conjecture? If not how > can we prove it? Well, if boring means non-trivial there are examples, namely opposites of cartesian closed categories. E.g. Set^op equivalent to CABA (complete atomic boolean algebras). E.g. for a topological space X the poset C(X) of closed subsets of X ordered by set inclusion is an example. There conegation ~A is the closure of the complement of A and A \cap ~A is the border of A. This received some attention as models of "dialectical logic". There are also biHeyting algebras meaning that A and A^op are Heyting algebras. Subobject lattices of objects in presheaf toposes are examples of this as observed by Lawvere. But if you mean by coexponential A x (_) having a left adjoint then in presence of a terminal object 1 this means A is isomorphic to 1 and indeed they are trivial. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]