From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2353 Path: news.gmane.org!not-for-mail From: "jpradines" Newsgroups: gmane.science.mathematics.categories Subject: Re: Function composition of natural transformations? (Pat Don... Date: Wed, 11 Jun 2003 10:37:35 +0200 Message-ID: <006901c32ff4$dc151240$a0d8f8c1@wanadoo.fr> References: <1d1.b5f6499.2c167d88@aol.com> Reply-To: "jpradines" NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018598 3725 80.91.229.2 (29 Apr 2009 15:23:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:18 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Thu Jun 12 13:02:13 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 12 Jun 2003 13:02:13 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19QUQu-0006L6-00 for categories-list@mta.ca; Thu, 12 Jun 2003 12:56:12 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 44 Original-Lines: 63 Xref: news.gmane.org gmane.science.mathematics.categories:2353 Archived-At: Thank you for your quick answer. I am not sure to understand it quite well. I must confess that I = consider myself much more as a differential geometer than as a = "categoricist", so that I am not at all well informed about categorical = literature. In fact I am often considered as a dissident in both fields, = since my general philosophy is : -first that there is a lot of interesting things to do in Geometry using = categories, an idea with which unfortunately most geometers, save = Ehresmann, disagree ; - secund that most of the classical tools developed by category = theorists, including Ehresmann, are not the best adapted for that = purpose, and I tried to develop some alternative tools (which indeed = most people ignore). It is not the place here to say more on that subject.=20 About the precise point in discussion the main object of my letter was = to emphasize the secund aspect of the definition (functor from I x A to = B or more symmetrically to I x B), which seems less familiar (is it = written somewhere ?) than the first one, and in my opinion the simplest = and the most useful (at least for my personal use). For each arrow f of = A from a to a', the image of I x {a, f, a'}, which may also be regarded = as an image of I x I, gives immediately a commutative square in A and = its diagonal (3 x 3 =3D 9 arrows including the unit ones), where one = reads immediately all the useful aspects of the data, displayed in a = perfectly geometric and symmetric way, in contrast with the usual = definitions, in which the two functors and the naturality arrow seem to = play quite unsymmetrical roles. (Of course all of this is so perfectly = obvious that I am convinced that plenty of people have noticed this, but = I must confess that I have never read (nor heared) it, though in my = opinion it is the best way of describing a natural transformation.)=20 I think also than it would be often useful to consider more frequently = the obvious canonical structure of double category on a product category = and especially on I x A, and I x I. Jean PRADINES ----- Original Message -----=20 From: Jpdonaly@aol.com=20 To: jpradines@wanadoo.fr=20 Cc: categories@mta.ca=20 Sent: Tuesday, June 10, 2003 2:17 AM Subject: Re: categories: Re: Function composition of natural = transformations? (Pat Don... 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.=20