From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7697 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology: Remarks Date: Fri, 3 May 2013 06:53:12 +0200 Message-ID: References: Reply-To: =?iso-8859-1?Q?Jean_B=E9nabou?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1283) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1367583895 27807 80.91.229.3 (3 May 2013 12:24:55 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 3 May 2013 12:24:55 +0000 (UTC) Cc: Categories To: Toby Bartels Original-X-From: majordomo@mlist.mta.ca Fri May 03 14:24:51 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 1UYF2f-00005Y-US for gsmc-categories@m.gmane.org; Fri, 03 May 2013 14:24:50 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44864) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UYEzs-00033z-HQ; Fri, 03 May 2013 09:21:56 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UYEzs-0003Ke-0E for categories-list@mlist.mta.ca; Fri, 03 May 2013 09:21:56 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7697 Archived-At: Dear Toby, I'm preparing an answer to all the mails I received about equivalence of = categories. In order to answer to yours, I need the following precisions = about your definition: (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? (ii) Let 1 denote the final category. The unique functor 1 -> 1 is the = unique equivalence between 1 and 1. How many spans 1 <- X -> 1 are = equivalences in your sense? Best regards, Jean Le 2 mai 2013 =E0 08:46, Toby Bartels a =E9crit : > Jean B?nabou wrote in small part: >=20 >> The one [notion of equivalence of categories] which might serve here = is f=20 >> full and faithful and essentially surjective. But unless we have AC = it is=20 >> not symmetric, even for A and B small. >=20 > Then the obvious thing to try is to symmetrise it: > An equivalence between A and B is a span A <- X -> B > of fully faithful and essentially surjective functors. >=20 >=20 > --Toby >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]