From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7699 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology: Remarks Date: Thu, 02 May 2013 22:41:38 -0300 Message-ID: References: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: plane.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 1367583962 28485 80.91.229.3 (3 May 2013 12:26:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 3 May 2013 12:26:02 +0000 (UTC) Cc: Categories To: Toby Bartels Original-X-From: majordomo@mlist.mta.ca Fri May 03 14:26:02 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 1UYF3o-0001J6-LW for gsmc-categories@m.gmane.org; Fri, 03 May 2013 14:26:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44878) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UYF2O-0003Sx-DK; Fri, 03 May 2013 09:24:32 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UYF2N-0003Ot-Lk for categories-list@mlist.mta.ca; Fri, 03 May 2013 09:24:31 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7699 Archived-At: On 02/05/13 03:46, Toby Bartels wrote: > Jean B?nabou wrote in small part: > >> The one [notion of equivalence of categories] which might serve here is f >> full and faithful and essentially surjective. But unless we have AC it is >> not symmetric, even for A and B small. > > 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. > > > --Toby > Equivalence of categories in practice is highly non symmetric. Usually one direction is defined and canonical, and the other is choice dependent and as such they are many of them. A definition of equivalence should reflect this fact, thus, it is not "a pair of functors such etc etc", but, either "A FUNCTOR full and faithful and essentially surjective", or "A FUNCTOR such that there exist a quasi inverse" if you do not want to use choice. e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]