From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8531 Path: news.gmane.org!not-for-mail From: Uwe Egbert Wolter Newsgroups: gmane.science.mathematics.categories Subject: Re: Category without objects Date: Fri, 06 Mar 2015 15:42:08 +0100 Message-ID: References: Reply-To: Uwe Egbert Wolter NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1425734204 14722 80.91.229.3 (7 Mar 2015 13:16:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 7 Mar 2015 13:16:44 +0000 (UTC) Cc: Categories list To: Peter LeFanu Lumsdaine Original-X-From: majordomo@mlist.mta.ca Sat Mar 07 14:16:36 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 1YUEap-0007b5-Cf for gsmc-categories@m.gmane.org; Sat, 07 Mar 2015 14:16:35 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:56514) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YUEZn-0001zo-Iv; Sat, 07 Mar 2015 09:15:31 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YUEZm-0006WS-RM for categories-list@mlist.mta.ca; Sat, 07 Mar 2015 09:15:30 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8531 Archived-At: Many thanks for all the immediate replies and all the interesting information. Finally, I could also reconstruct today where I have seen the arrows-only definition around 30 years ago. There is a four page introduction into categories in the first chapter of P.M. Cohn's "Universal Algebra". He outlines that one could do so and gives a corresponding exercise. Best Uwe On 2015-03-06 00:45, Peter LeFanu Lumsdaine wrote: > We actually had a post-seminar reference-hunt on this in Stockholm > quite recently, and found that the arrows-only definition goes right > back to Mac Lane 1948, ???Groups, Categories, and Duality???: > http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079106/pdf/pnas01707-0037.pdf > > > This cites two earlier papers only along with the definition (Mac Lane > 1942 and Eilenberg???Mac Lane 1945 ??? the first two papers to mention > categories, right?), but both of those used the objects-and-arrows > formulation. So it seems that the two-sorted formulation was > considered right from the start, and the arrows-only version either > from the start or very soon afterwards. > > Of course, the original question has already been well answered, but I > guess the extra history may be of interest to others as well. > > Best, > ???Peter. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]