From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2313 Path: news.gmane.org!not-for-mail From: Marc Olschok Newsgroups: gmane.science.mathematics.categories Subject: Re: Function composition of natural transformations? Date: Mon, 2 Jun 2003 16:14:16 +0200 (MESZ) Message-ID: <200306021414.QAA18084@d2-hrz.uni-duisburg.de> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018567 3503 80.91.229.2 (29 Apr 2009 15:22:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:47 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jun 2 17:36:32 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 02 Jun 2003 17:36:32 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19Mw2G-0004mi-00 for categories-list@mta.ca; Mon, 02 Jun 2003 17:36:04 -0300 X-Scanned-By: MIMEDefang 2.21 (www . roaringpenguin . com / mimedefang) Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 3 Original-Lines: 40 Xref: news.gmane.org gmane.science.mathematics.categories:2313 Archived-At: > Here is a technical/pedagogical question which has been bothering me for > about twelve years. > > In problem 5 on page 19 of "Categories for the Working Mathematician" (CWM), > Saunders Mac Lane points out that a natural transformation may be fully > extended to an intertwining function from one category to another, > intertwining meaning (except in the void case), that the function > transforms on one side according to its domain functor and on the > other according to its codomain functor. > Then on page 42 Mac Lane introduces what he calls "horizontal" composition > diagramatically and without reference to the fully extended intertwining > functions. But the function composite of such a pair of functions trivially > intertwines the function composite of the domain functors with that of > the codomain functors, and this function composition operation is very > quickly verified to be "horizontal" composition when written in terms > of restrictions to sets of objects. Thus Mac Lane and everyone else I > have read leaves the impression that an honest verification of, say, > the associativity of "horizontal" composition would require a somewhat > involved diagrammatic demonstration which, in fact, would be nothing > other than the hard way to prove the associativity of function composition. > Presumably this has been noticed for a long, long time, but the > 1998 edition of CWM did not mention it, and I can't help but be struck > by the fact that other authors' terminologies leave the impression that > they don't know or don't care that "horizontal", star or Godement > composition is function composition.[...] At least in the book "Elemente der Kategorientheorie" by D. Pumpl\"un the above characterization of natural maps is used explicitely; there is also a short discussion on obtaining simpler proofs this way. For the above reason \circ is used for the "horizontal composition"; \cdot or \ast (I do not remember which one) is used for the "vertical composition", which after all looks more "point-wise". Unfortunately some authors use these symbols just the other way round. Marc