From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2344 Path: news.gmane.org!not-for-mail From: "jpradines" Newsgroups: gmane.science.mathematics.categories Subject: Re: Function composition of natural transformations? (Pat Donaly) Date: Mon, 9 Jun 2003 00:03:06 +0200 Message-ID: <003001c32e09$ddf7e520$af51f8c1@wanadoo.fr> References: <1c5.9c17e04.2c097945@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 1241018593 3687 80.91.229.2 (29 Apr 2009 15:23:13 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:13 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Mon Jun 9 11:41:33 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 09 Jun 2003 11:41:33 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19PNp4-0002jE-00 for categories-list@mta.ca; Mon, 09 Jun 2003 11:40:34 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 35 Original-Lines: 66 Xref: news.gmane.org gmane.science.mathematics.categories:2344 Archived-At: Just a few naive remarks about natural transformations and their two = compositions. I think comfortable to define natural transformations as functors, which = is not the case, neither for the classical "object to arrow" nor for the = "arrow to arrow" definition. Now this can be achieved in two obviously equivalent ways. In the following the letter I (though or because it is frequently used = also to denote the unit interval) will picture the category denoted by 2 = in CWM (just one arrow e going from the object 0 to the object 1) while = QA will denote the category of commutative squares in the category A, = endowed with the "horizontal" composition. (Note that the arrows of A = may be described as functors from I to A and the arrows of QA as = functors from the product category I x I to A). Then the first definition is : a natural transformation between the = categories A and B is nothing else than a functor from A to QB.=20 With this first definition, the vertical composition of natural = transformations comes immediately from the "vertical" composition in QB = (which is indeed a double category). The secund definition (which is just a more symmetric reformulation of = the "arrow to arrow" definition) seems to be less popular, though very = useful in my opinion : a natural transformation between the categories A = and B is also nothing else than a functor from the product category I x = A to B.=20 This can also may be viewed as a functor from I x A to I x B, letting = the first component be just the canonical projection from I x A to I. Then the horizontal composition is just the composition of functors. The latter definition is especially convenient when working with = categories (or groupoids) in a category. For instance one knows = immediately what is a smooth natural transformation (of course I is = endowed with the discrete structure). It also allows to define = immediately the horizontal composition for "higher order natural = transformations", replacing I by its n'th power. Jean PRADINES ----- Original Message -----=20 From: To: Sent: Saturday, May 31, 2003 5:19 AM Subject: categories: Function composition of natural transformations? = (Pat Donaly) > Here is a technical/pedagogical question which has been bothering me = for > about twelve years. >=20 > 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.