From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7041 Path: news.gmane.org!not-for-mail From: Uwe.Wolter@ii.uib.no Newsgroups: gmane.science.mathematics.categories Subject: Re: The boringness of the dual of exponential Date: Wed, 09 Nov 2011 21:57:23 +0100 Message-ID: References: Reply-To: Uwe.Wolter@ii.uib.no NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; DelSp="Yes"; format="flowed" Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1320886162 31014 80.91.229.12 (10 Nov 2011 00:49:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 10 Nov 2011 00:49:22 +0000 (UTC) Cc: David Leduc , categories@mta.ca To: Paul Taylor Original-X-From: majordomo@mlist.mta.ca Thu Nov 10 01:49:17 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 1ROIpR-0004SF-Cc for gsmc-categories@m.gmane.org; Thu, 10 Nov 2011 01:49:17 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:54625) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1ROIoF-0006uG-6o; Wed, 09 Nov 2011 20:48:03 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ROIoD-0003SR-It for categories-list@mlist.mta.ca; Wed, 09 Nov 2011 20:48:01 -0400 In-Reply-To: Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7041 Archived-At: Some short remarks from a dilettante in universal algebra, category theory and philosophy: I came along the unsatisfactory asymmetry in the category SET when looking at coalgebras and asking about coequations some years ago. Having in mind that kernels play a central role in universal algebra I was expecting that co-kernels should play a similar role in universal coalgebra. One can easily dualize the kernel reasoning. Such a dualization, however, looks kind of artificial since a co-kernel just provides a "strange coding" of the image of a map, i.e., a subset, so to speak. One can identify the asymmetry by looking at exponentiation, but isn't the asymmetry already related to the fact that SET is a distributive category, i.e., that addition is trivial and boring compared to multiplication? My interest in coalgebras was also a kind of philosophically triggered. The "classical western scientific cultur" relies and focusses on the existence of "objects/things" and the category SET is somehow the abstract essence of this perception of the world as a bunch of objects/things which are assumed to be identifiable and existing until the end of the time. Buddhistic and/or dialectical reasoning, in contrast, perceives the world as a net of mutual dependent and interweaved "processes". So, my question was if there is anything in mathematics reflecting on a formal level this buddhistic and dialectical reasoning based on "processes". I don't consider coalgebras in SET as such formalization. Those coalgebras are based on "object/thing reasoning" thus they can only give an approximation of the philosophical concept of a "process", namely in terms of distinctions and observations. Uwe Wolter Quoting Paul Taylor : > 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/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]