From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7706 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology: Remarks Date: Fri, 3 May 2013 22:34:17 -0700 Message-ID: <20130504053417.GA7450@ugcs.caltech.edu> References: Reply-To: Toby Bartels NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1367675115 23795 80.91.229.3 (4 May 2013 13:45:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 4 May 2013 13:45:15 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Sat May 04 15:45:15 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UYcm3-0004OO-3Y for gsmc-categories@m.gmane.org; Sat, 04 May 2013 15:45:15 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45417) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UYcka-0000Wi-TZ; Sat, 04 May 2013 10:43:44 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UYcka-0000nA-GZ for categories-list@mlist.mta.ca; Sat, 04 May 2013 10:43:44 -0300 Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7706 Archived-At: Jean B?nabou wrote in part: >(i) If F: A -> B and G: B -> C are full and faithful essentially surjective functors, so is GF. How do you compose your equivalences? Good question! In this case, we can compose by pullback. But when I wrote "the obvious thing to do", maybe I should have written "one obvious thing to try". After all, it might not work; in this case, it does. If it didn't work, another obvious thing to try would be zigzags. In this case, this gives an equivalent 2-groupoid. So in principle, one could do either, but spans are simpler. Generalising from the 2-groupoid of categories to the 2-category of them, we can use spans A <- X -> B where only A <- X needs to be ff eso. This is what Michael Makkai did, and it allows one to avoid AC while retaining the usual results about the 2-category of categories. (Pace Tom Leinster's recent comment under this thread, Makkai actually required A <- X to be strictly surjective on objects, but again this does not matter; the resulting 2-category is equivalent.) Beyond this, let me just say that I agree with Marta's answers. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]