From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2351 Path: news.gmane.org!not-for-mail From: Jpdonaly@aol.com Newsgroups: gmane.science.mathematics.categories Subject: Re: Function composition of natural transformations? (Pat Don... Date: Mon, 9 Jun 2003 20:17:12 EDT Message-ID: <1d1.b5f6499.2c167d88@aol.com> 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 1241018597 3719 80.91.229.2 (29 Apr 2009 15:23:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:17 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Jun 10 19:16:36 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 10 Jun 2003 19:16:36 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19PrP1-00053w-00 for categories-list@mta.ca; Tue, 10 Jun 2003 19:15:39 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 42 Original-Lines: 18 Xref: news.gmane.org gmane.science.mathematics.categories:2351 Archived-At: Thanks, Jean. As an assiduous student of CWM, I was aware of this and will always wonder why Mac Lane didn't just make the point explicit in his first edition in 1971. The only thing left to realize is that the category of commutative squares which you mention is a subcategory of a product category and thus has a couple of projection functors on it which can be used to follow a functor to get the domain and codomain functors of the natural transformation, so that this version of naturality is much more neatly packaged than the usual diagram. I believe that there is a worker named John Baez (deep apologies for any naive and unforgivable errors here) who says that Mac Lane claimed to be interested not in functoriality so much as naturality when he was coinventing category theory; I wonder when and if he realized that naturality is a brand of functoriality. It would seem that this realization would come very early. In general, if one fixes an argument in a bifunctor, the resulting function is a fully extended intertwining function, and I believe that your point is that every natural transformation arises in this way. So already naturality is an artifact of functoriality. Mitchell notices much of this in his 1965 book.