From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7705 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology: Remarks Date: Fri, 3 May 2013 19:22:12 -0400 Message-ID: References: ,<16016_1367583941_5183ACC5_16016_35_1_E1UYF2N-0003Ot-Lk@mlist.mta.ca> Reply-To: Marta Bunge NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1367675050 23213 80.91.229.3 (4 May 2013 13:44:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 4 May 2013 13:44:10 +0000 (UTC) Cc: "categories@mta.ca" To: E D , Toby Bartels Original-X-From: majordomo@mlist.mta.ca Sat May 04 15:44:10 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 1UYckz-0003TE-GK for gsmc-categories@m.gmane.org; Sat, 04 May 2013 15:44:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45411) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UYcjk-0000K4-GD; Sat, 04 May 2013 10:42:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UYcjk-0000kZ-2X for categories-list@mlist.mta.ca; Sat, 04 May 2013 10:42:52 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7705 Archived-At: Dear Eduardo=2C=0A= =0A= This is an addendum to what I have written to Jean Benabou (and Toby Bartel= s)=2C but referring to your comment. It is not always the case that an equi= valence F: S^G -> S^K=2C for S a topos=2C and G=2CK small (internal)=A0grou= pies=2C is induced by an equivalence f:G->K=2C not even by a weak equivalen= ce functor f:G->K. However=2C given that S^G and S^G are equivalent categor= ies=2C it follows that the stack completions G' and K' are equivalent. In p= articular=2C this (the Morita=A0equivalence theorem for internal=A0groupoid= s=A0in a topos) is an instance where it is necessary to consider=2C not jus= t wef G->K or K->G=2C but the equivalence relation "G weakly equivalent to = K".=A0=0A= =0A= Regards=2C=0A= Marta=0A= =0A= =0A= =0A= =0A= =0A= =0A= =0A= > Date: Thu=2C 2 May 2013 22:41:38 -0300=0A= > From: edubuc@dm.uba.ar=0A= > To: categories@TobyBartels.name=0A= > CC: categories@mta.ca=0A= > Subject: categories: Re: Terminology: Remarks=0A= > =0A= > On 02/05/13 03:46=2C Toby Bartels wrote:=0A= >> Jean B?nabou wrote in small part:=0A= >>=0A= >>> The one [notion of equivalence of categories] which might serve here i= s f=0A= >>> full and faithful and essentially surjective. But unless we have AC it= is=0A= >>> not symmetric=2C even for A and B small.=0A= >>=0A= >> Then the obvious thing to try is to symmetrise it:=0A= >> An equivalence between A and B is a span A<- X -> B=0A= >> of fully faithful and essentially surjective functors.=0A= >>=0A= >>=0A= >> --Toby=0A= >>=0A= > =0A= > Equivalence of categories in practice is highly non symmetric. Usually=0A= > one direction is defined and canonical=2C and the other is choice=0A= > dependent and as such they are many of them. A definition of equivalence= =0A= > should reflect this fact=2C thus=2C it is not "a pair of functors such et= c=0A= > etc"=2C but=2C either "A FUNCTOR full and faithful and essentially=0A= > surjective"=2C or "A FUNCTOR such that there exist a quasi inverse" if yo= u=0A= > do not want to use choice.=0A= > =0A= > e.d.=0A= > =0A= > =0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]