From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7045 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: The boringness of the dual of exponential Date: Thu, 10 Nov 2011 09:29:06 +0000 (GMT) Message-ID: References: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1320954782 27062 80.91.229.12 (10 Nov 2011 19:53:02 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 10 Nov 2011 19:53:02 +0000 (UTC) Cc: David Leduc , categories@mta.ca To: Paul Taylor Original-X-From: majordomo@mlist.mta.ca Thu Nov 10 20:52:58 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 1ROag9-0003st-5o for gsmc-categories@m.gmane.org; Thu, 10 Nov 2011 20:52:53 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:55014) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1ROae0-0007Fe-Vl; Thu, 10 Nov 2011 15:50:40 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ROadz-0006jj-9H for categories-list@mlist.mta.ca; Thu, 10 Nov 2011 15:50:39 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7045 Archived-At: One point that no-one has mentioned yet is that you can't have exponentiation and its dual in the same category, unless it is a preorder. If exponentiation exists, then the initial object 0 is strict, and so 0 x 0 = 0 (read all equality signs as isomorphisms). But if A + (-) has a left adjoint then it distributes over product, so A = A + 0 = A + (0 x 0) = (A + 0) x (A + 0) = A x A which implies that any two maps into A (with the same domain) are equal. Of course, bi-Heyting algebras (posets P such that both P and P^op are cartesian closed) are of some interest, as has already been mentioned; but if you want to work in non-preordered categories then you have to choose one or the other. Peter Johnstone On Tue, 8 Nov 2011, Paul Taylor wrote: > When David originally posted his question, I thought it was rather > a silly one and that it was quite rightly dismissed by various > people. On the other hand, he now says > >> 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? > > In other words, these things are "idioms" or "naturally occurring > things" in mathematics, but there is a gap in the obvious symmetries. > > Looking for gaps in symmetries is a good thing to do. For example > Dirac (whose biography by Graham Farmelo I have just started reading) > predicted the positron this way. > > Actually, if we're looking at the categorical structure of the category > of sets, it isn't very symmetrical at all. The second edition of > Paul Cohn's "Universal Algebra" was evidently influenced by Mac Lane's > famous textbook, but illustrates how categorists had way overemphasised > duality. > > For example the terminal object yields the classical notion of element > or point, whereas the initial object is strict and boring. > > Products and coproducts of sets are very different. > > I explored this kind of thing in my book. For example, the section > on coproducts shows how different they are in sets/spaces and algebras. > > So David's question becomes a good one that deserves an answer if > we read it as one about the phenomenology of mathematics rather than > its technicalities. > > Paul Taylor > > PS There is a boring technical answer that I don't think anyone has > mentioned, namely copowers, especially of modules. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]