From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2309 Path: news.gmane.org!not-for-mail From: Jpdonaly@aol.com Newsgroups: gmane.science.mathematics.categories Subject: Function composition of natural transformations? (Pat Donaly) Date: Fri, 30 May 2003 23:19:33 EDT Message-ID: <1c5.9c17e04.2c097945@aol.com> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 X-Trace: ger.gmane.org 1241018564 3482 80.91.229.2 (29 Apr 2009 15:22:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:44 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jun 2 10:17:48 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 02 Jun 2003 10:17:48 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19Mp7M-0004FY-00 for categories-list@mta.ca; Mon, 02 Jun 2003 10:12:52 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 59 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:2309 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. Notationally, I am bemused to see the standard symbol for function composition (the small circle) degraded into a generic symbol for the composition of just about any category, while a star or an asterisk is frequently used to denote what amounts to function composition done awkwardly. This worries me that I am somehow overlooking something fairly blatant. Can someone tell me what it is? Jpdonaly@aol.com