From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7035 Path: news.gmane.org!not-for-mail From: "Paul Taylor" Newsgroups: gmane.science.mathematics.categories Subject: The boringness of the dual of exponential Date: Tue, 8 Nov 2011 16:20:06 -0000 Message-ID: References: Reply-To: "Paul Taylor" 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 1320847836 13535 80.91.229.12 (9 Nov 2011 14:10:36 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 9 Nov 2011 14:10:36 +0000 (UTC) To: "David Leduc" , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed Nov 09 15:10:28 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 1RO8rC-0006Sb-UB for gsmc-categories@m.gmane.org; Wed, 09 Nov 2011 15:10:27 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:53262) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RO8oy-00080T-Rw; Wed, 09 Nov 2011 10:08:08 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RO8ow-0000Zd-TT for categories-list@mlist.mta.ca; Wed, 09 Nov 2011 10:08:06 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7035 Archived-At: 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/ ]