From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4549 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Categories and functors, query Date: Mon, 8 Sep 2008 08:50:41 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241020020 13799 80.91.229.2 (29 Apr 2009 15:47:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:00 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Sep 8 20:52:27 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Sep 2008 20:52:27 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcqRi-00021M-Rj for categories-list@mta.ca; Mon, 08 Sep 2008 20:47:02 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 18 Original-Lines: 118 Xref: news.gmane.org gmane.science.mathematics.categories:4549 Archived-At: Interesting speculation, but how can we verify or refute it? What I can add is that when I sat in on Sammy's category theory course (called homological algebra, but I am not sure Ext or Tor were ever mentioned), I do not recall that he so much as mentioned groupoids. I once mentioned to Charles Ehresmann that he appeared to view categories as a generalization of groupoids while Eilenberg and Mac Lane thought of them as a generalization of posets. Charles agreed. This reminds me of a speculation I have often had (although Saunders denied and he knew Birkhoff pretty well). In the 30s and 40s, the word "homomorphism" was regularly used but always meant surjective. By the late 40s and 50s people were talking about "homomorphism into" meaning not necessarily surjective. So groups had lattices of subgroups and lattices of quotient groups and Birkhoff invented lattice theory at least partly in the hope that the structure of those two lattices would tell you a lot about the structure of the group. I don't think this actually happened to any great extent. But I have wondered whether Birkhoff might instead have invented categories had our more general notion of homomorphism been rampant. As I said Saunders didn't think so, but it still sounds attractive to me. One of the things that astonishes me about "General theory of natural equivalences" is that they clearly knew about natural transformations in general but chose to talk only about equivalences. I once asked Sammy about that and he more or less said something like one generalization at a time. But they must have realized that the Hurevic map is a superior example. Still, Steenrod must have gotten the point immediately. Michael On Sun, 7 Sep 2008, R Brown wrote: > There is another curiosity about the axioms for a category, namely the > infuence of the known axioms for a groupoid (Brandt, 1926). Bill Cockcroft > told me that these axioms had influenced E-M. These axioms were well used in > the algebra group at Chicago. However when I asked Sammy about this in 1985 > he firmly said `no, and was why the notion of groupoid did not appear as an > example in the E-M paper'! > > Perhaps it was a case of forgetting the influence? > > Ronnie > > > > > ----- Original Message ----- > From: "Johannes Huebschmann" > To: > Sent: Saturday, September 06, 2008 11:48 AM > Subject: categories: Categories and functors, query > > >> Dear All >> >> I somewhat recall that, a while ago, we discussed the origins of >> the notions of category and functor. S. Mac Lane had once pointed >> out to me these origins but from my recollections we did not >> entirely reproduce them. >> >> In his paper >> >> Samuel Eilenberg and Categories, JPAA 168 (2002), 127-131 >> >> Saunders Mac Lane clearly pointed out the origins: >> >> "Category" from Kant (which I had known all the time) >> >> "Functor" from Carnap's book "Logical Syntax of Language" (which I >> had forgotten). >> >> >> Also I have a question, not directly related to the above issue: >> >> I have seen, on some web page, a copy of >> the referee's report about the Eilenberg-Mac Lane paper >> where Eilenberg-Mac Lane spaces are introduced. >> I cannot find this web page (or the report) >> any more. Can anyone provide me with >> a hint where I can possibly find it? >> >> Many thanks in advance >> >> Johannes >> >> >> >> HUEBSCHMANN Johannes >> Professeur de Mathematiques >> USTL, UFR de Mathematiques >> UMR 8524 Laboratoire Paul Painleve >> F-59 655 Villeneuve d'Ascq Cedex France >> http://math.univ-lille1.fr/~huebschm >> >> TEL. (33) 3 20 43 41 97 >> (33) 3 20 43 42 33 (secretariat) >> (33) 3 20 43 48 50 (secretariat) >> Fax (33) 3 20 43 43 02 >> >> e-mail Johannes.Huebschmann@math.univ-lille1.fr >> >> >> > > > -------------------------------------------------------------------------------- > > > > No virus found in this incoming message. > Checked by AVG - http://www.avg.com > Version: 8.0.169 / Virus Database: 270.6.17/1657 - Release Date: 06/09/2008 > 20:07 > > >