From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2746 Path: news.gmane.org!not-for-mail From: Max Kelly Newsgroups: gmane.science.mathematics.categories Subject: Dinatural transformations Date: Mon, 05 Jul 2004 16:35:33 +1000 Message-ID: <40E8F6B5.9010406@usyd.edu.au> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018869 5596 80.91.229.2 (29 Apr 2009 15:27:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:27:49 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jul 5 16:08:32 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Jul 2004 16:08:32 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BhYoG-0007Oc-00 for categories-list@mta.ca; Mon, 05 Jul 2004 16:07:24 -0300 User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.0; en-US; rv:1.4) Gecko/20030624 Netscape/7.1 (ax) X-Accept-Language: en-us, en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 9 Original-Lines: 29 Xref: news.gmane.org gmane.science.mathematics.categories:2746 Archived-At: Nason Yanofsky asked a question about composition of dinatural transformations, and there have been a number of replies - but none giving the reference I should have expected, namely [S.Eilenberg and G.M.Kelly, A generalization of the functorial calculus,, J.Algebra 3 (1966), 366 - 375] . What Sammy and I were concerned with were such families as the evaluationn e_A,B : [A,B] o A --> B , where o is a tensor product and [ , ] is an internal hom. Here e_A,B is natural in B in the usual sense; it is also natural, in our extended sense, in the variable A, which appears twice on one side of the arrow, but with two opposite variances. Similar for d_A,B : A --> [B, AoB]. In certain circumstances one can compose such "natural transformations" to form new ones:one example is the composite AoB ---------> [B, AoB] o B ----------> AoB d_A,B o B e_B, AoB which is in fact the identity natural transformation. Sammy and I gave a general treatment in the article above. Later, others generalised our "extended naturals" to get the notion of dinatural transformation. Since these do not compose except in some very special cases, I find them to be of limited interest. In contrast, I find that I still use the Eilenberg-Kellycalculus from time to time. Max Kelly.