From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7039 Path: news.gmane.org!not-for-mail From: "Reinhard Boerger" Newsgroups: gmane.science.mathematics.categories Subject: Re: The boringness of the dual of exponential Date: Wed, 9 Nov 2011 10:19:30 +0100 Message-ID: Reply-To: "Reinhard Boerger" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1320848132 16169 80.91.229.12 (9 Nov 2011 14:15:32 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 9 Nov 2011 14:15:32 +0000 (UTC) Cc: To: "'David Leduc'" Original-X-From: majordomo@mlist.mta.ca Wed Nov 09 15:15:28 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 1RO8w2-0000He-5M for gsmc-categories@m.gmane.org; Wed, 09 Nov 2011 15:15:26 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:53274) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RO8ql-00087p-GQ; Wed, 09 Nov 2011 10:09:59 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RO8qi-0000bQ-Hi for categories-list@mlist.mta.ca; Wed, 09 Nov 2011 10:09:56 -0400 Thread-Index: AcyeL0xwVEZuNekER9eAnqpbk4IjjwAj4Hsg Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7039 Archived-At: Hello! David Leduc is not satisfied: >> Take any category E where exponentiable is interesting. >> Then the dual of exponentiable is not boring in E^op. >=20 > Indeed! And this is clearly true of the example given by Thomas, = namely > Set^op. >=20 > 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? I wonder whether it makes sense to introduce notions, which only in the duals of familiar categories. Of course, Set^op is equivalent to the category of complete atomic Boolean algebras, but I do not see that the = dual of exponentiation plays an important role in the theory if these Boolean algebras. > 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 In classical logic, one can form this "co-implication" but it does not = look very interesting to me. In intuitionistic logic I do not see how to add = it more ore less meaningfully (e.g. in such a way that it is left adjoint = to "or" in the first argument). Greetings Reinhard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]