From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8534 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: Category without objects Date: Sun, 08 Mar 2015 13:44:22 -0300 Message-ID: References: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1425856283 27188 80.91.229.3 (8 Mar 2015 23:11:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 8 Mar 2015 23:11:23 +0000 (UTC) Cc: Categories list To: Ronnie Brown , Original-X-From: majordomo@mlist.mta.ca Mon Mar 09 00:11:16 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1YUkLr-0008Fm-U5 for gsmc-categories@m.gmane.org; Mon, 09 Mar 2015 00:11:16 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58877) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YUkKi-00056I-2e; Sun, 08 Mar 2015 20:10:04 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YUkKi-0004TQ-3a for categories-list@mlist.mta.ca; Sun, 08 Mar 2015 20:10:04 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8534 Archived-At: It seems to me: Eheresmann arrived to the concept of categories as a generalization of groupoids, and he was dealing with small internal categories (the set of arrows (or elements) were differential manifolds etc). This explains why he dismissed objects in his later treatment of abstract categories. Eilemberg and MacLane arrived to the concept of categories as an abstraction of the large concrete categories of sets with structure and functions which were considered to be morphisms for the structure. Objects were essential in this approach. That the insight of E. M. to do not dismiss objects in the abstract setting was wonderful is that to-day we can not conceive groupoids without objects. On 7/3/15 11:36, Ronnie Brown wrote: > I remember Henry Whitehead said that he was very impressed by the axioms > for a category in the Eilenberg-Mac Lane paper. > > A curiosity about the definition is that groupoids were defined by > Brandy in 1926, and this definition was used by the Chicago school of > algebra and applied to ring theory. Bill Cockcroft told me that the > groupoid notion was an influence. In 1985 I asked Eilenberg about this, > and said no, since if it had been, they would have used it as an > example! I forgot to ask Mac Lane! > > Ronnie Brown > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]