From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4550 Path: news.gmane.org!not-for-mail From: Walter Tholen Newsgroups: gmane.science.mathematics.categories Subject: Re: Categories and functors, query Date: Mon, 08 Sep 2008 12:00:58 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020020 13803 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 20:53:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Sep 2008 20:53:52 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcqT9-00025Z-6Y for categories-list@mta.ca; Mon, 08 Sep 2008 20:48:31 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 19 Original-Lines: 49 Xref: news.gmane.org gmane.science.mathematics.categories:4550 Archived-At: There is another aspect to the E-M achievement that I stressed in my CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the extent to which 20th-century mathematics was entrenched in set theory, it was a tremendous psychological step to put structure on "classes" and to dare regarding these (perceived) monsters as objects that one could study just as one would study individual groups or topological spaces. In my experience, skepticism towards category theory is often rooted in the fear of the "illegitimately large" size, till today. By comparison, Brandt groupoids lived in the cozy and familiar small world, and their definition was arrived at without having to leave the universe. With the definition of category (and functor and natural transformation) Eilenberg and Moore had to do a lot more than just repeating at the monoid level what Brandt did at the group level! In my view their big psychological step here is comparable to Cantor's daring to think that there could be different levels of infinity. Cheers, Walter. Dana Scott wrote: > > On Sep 7, 2008, at 2:33 PM, 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? > > > I certainly heard Saunders mention Brandt groupoids as examples. > (Not very good examples, since all maps are invertible.) But, as > everyone knows, it is not the definition of a category that > is the key part, but seeing that functors and natural > transformations are interesting. > >