From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7032 Path: news.gmane.org!not-for-mail From: David Leduc Newsgroups: gmane.science.mathematics.categories Subject: Re: The boringness of the dual of exponential Date: Mon, 7 Nov 2011 12:52:07 +0000 Message-ID: References: Reply-To: David Leduc NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1320767999 644 80.91.229.12 (8 Nov 2011 15:59:59 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 8 Nov 2011 15:59:59 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Tue Nov 08 16:59:55 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 1RNo5Z-00054q-3Z for gsmc-categories@m.gmane.org; Tue, 08 Nov 2011 16:59:53 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:48187) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RNo1T-0004Wd-K9; Tue, 08 Nov 2011 11:55:39 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RNo1R-00088Q-MC for categories-list@mlist.mta.ca; Tue, 08 Nov 2011 11:55:37 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7032 Archived-At: On Mon, Nov 7, 2011 at 07:59, Ross Street wrote: > The conjecture is false. > Take any category E where exponentiable is interesting. > Then the dual of exponentiable is not boring in E^op. Indeed! And this is clearly true of the example given by Thomas, namely Set^op. However, I am not yet satisfied. Let me precise my thoughts. In the textbooks and lecture notes on category category that I have read, there are always product and coproduct, pullback and pushout, equalizer and coequalizer, monomorphism and epimorphism, and so on. However exponential is always left alone. That is why I assumed it is boring. If it is not boring, why is it never mentioned in textbooks and lecture notes on category theory? Also, in logic, "and" goes in pair with "or", "for all" goes in pair with "there exists". But implication is always left alone. Why is it so? Is it not the case in "dialectical logic" mentioned by Thomas? By the way, I'd love to have some reference on models of dialectical logic. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]