From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7034 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Re: The boringness of the dual of exponential Date: Mon, 7 Nov 2011 11:32:31 -0500 Message-ID: References: , Reply-To: "F. William Lawvere" 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 1320768064 1324 80.91.229.12 (8 Nov 2011 16:01:04 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 8 Nov 2011 16:01:04 +0000 (UTC) Cc: categories To: THOMAS STREICHER , david leduc Original-X-From: majordomo@mlist.mta.ca Tue Nov 08 17:01:00 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 1RNo6d-0005e0-KL for gsmc-categories@m.gmane.org; Tue, 08 Nov 2011 17:01:00 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:36614) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RNo3R-0004fU-PG; Tue, 08 Nov 2011 11:57:41 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RNo3P-0008AB-Vf for categories-list@mlist.mta.ca; Tue, 08 Nov 2011 11:57:39 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7034 Archived-At: The lattice of all subtoposes of any given toposis surely not boring.For ex= ample the lattice of positive model classes of=20 a given theory needs investigating. Bill > Date: Sun=2C 6 Nov 2011 22:55:08 +0100 > From: streicher@mathematik.tu-darmstadt.de > To: david.leduc6@googlemail.com > CC: categories@mta.ca > Subject: categories: Re: The boringness of the dual of exponential >=20 >> 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? >=20 > Well=2C if boring means non-trivial there are examples=2C namely opposite= s > 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. >=20 > Thomas >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]