From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7701 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories,gmane.spam.detected Subject: Re: Terminology: Remarks Date: Fri, 3 May 2013 11:07:37 -0400 Message-ID: References: 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 1367674803 21018 80.91.229.3 (4 May 2013 13:40:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 4 May 2013 13:40:03 +0000 (UTC) To: Toby Bartels , , Jean Benabou Original-X-From: majordomo@mlist.mta.ca Sat May 04 15:40:03 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 1UYcgw-0007zj-Ub for gsmc-categories@m.gmane.org; Sat, 04 May 2013 15:39:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45381) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UYcet-0008EX-04; Sat, 04 May 2013 10:37:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UYces-0000dy-IB for categories-list@mlist.mta.ca; Sat, 04 May 2013 10:37:50 -0300 Precedence: bulk X-Spam-Report: 7.2 points; * 1.8 DATE_IN_PAST_12_24 Date: is 12 to 24 hours before Received: date * 2.5 LOCALPART_IN_SUBJECT Local part of To: address appears in Subject * 1.1 FORGED_HOTMAIL_RCVD2 hotmail.com 'From' address, but no 'Received:' * 1.8 MIME_QP_LONG_LINE RAW: Quoted-printable line longer than 76 chars Xref: news.gmane.org gmane.science.mathematics.categories:7701 gmane.spam.detected:5055936 Archived-At: Dear Jean (and Toby)=2C=A0=0A= =0A= =0A= The notion of equivalence of categories suggested by Toby is precisely the = one introduced in the paper Marta Bunge and Robert Pare=2C "Stacks and Equi= valence of Indexed Categories"=2C Cahiers de Top. et Geo. Diff. =2C Vol XX-= 4 (1979) 373-399. =A0S is a topos.=A0=0A= =0A= Definition (1.3) =A0We say that two S-indexed categories A and B are weakly= equivalent if there exists an S-indexed category E and a span A <-E->B of = weak equivalence functors (fully faithful and essentially surjective) E->A = and E->B.=0A= =0A= Proposition (1.4) Weak equivalence is an equivalence relation. Proof. The r= elation is trivially reflexive and symmetric. Composition is defined by mea= ns of (what we call) a 2-pullback=2C from which it follows that the relatio= n is transitive.=A0=0A= =0A= This answers your question (i).=0A= =0A= =0A= Proposition (1.6) Let C and D be small categories (internal to S). If =A0th= eir externalizations [C] and [D] are weakly equivalent=2C then one can choo= se D small with [D] ->[C]=2C[D] -> [B] weak equivalences. In this case one = can simply resort to entirely internal definition of weak equivalence so th= at =A0C <-D->B is a span in S. Proof. See pages 377-378.=A0=0A= =0A= Remark (page 384). Equivalence implies weak equivalence. Weak equivalence i= mplies equivalence iff (AC) holds. Weak equivalence implies local equivalen= ce iff (IAC) holds Local equivalence does not imply weak equivalence. Equiv= alence implies local equivalence. Local equivalence implies equivalence iff= every object of S with full support has a global section.=A0=0A= =0A= Corollary (2.12) (to Definition 2.10). Let F:A->B be a weak equivalence fun= ctor between S-indexed categories where B is a stack. Then F:A->B is "the" = (that is=2C up to equivalence) stack completion of A.=0A= =0A= Corollary (2.12). Let A and B be S-indexed categories and let F: A->A'=2C G= :B->B' be weak equivalence functors with A' and B' stacks. If A and B are w= eakly equivalent=2C then A' and B' are equivalent=2C and conversely.=A0=0A= =0A= Rekark (at the end of Section 2). The associated stack of any locally small= S-indexed category is constructed in (Marta Bunge=2C Stack completions and= Morita equivalence for categories in a topos=2C Cahiers de Top.Gem. Diff = =A0XX-4 (1979) 401-436). The stack completion of a locally small category A= is (can be taken to be) a locally small category A'. However=2C it is not = in genera the case that the stack completion of a small category C =A0be sm= all. Say that S satisfies the Axiom of Stack Completions (ASC) if the stack= completion of a category in S can be taken to be small. Any Grothendieck t= opos S satisfies (ASC).=0A= =0A= =0A= Let me point out that nothing special about S-indexed categories is employe= d here that could not have been done with fibrations over S. We could have = defined weak equivalence of small categories (in S) directly but the genera= l theory of stacks demanded that we dealt with arbitrary S-indexed catgeori= es=2C or with arbitrary fibrations over S.=A0=0A= =0A= =0A= As for an answer to your question (ii)=2C any small category X with X->1 a = weak equivalence gives a span 1<-X->1 as desired. One of them is of course = 1<-1->1. Such categories X have 1 as their stack completion=2C so they are = all themselves weakly equivalent. In other words=2C up to weak equivalence = there is only one such span 1<-X->1.=A0=0A= =0A= =0A= Best regards=2C=0A= Marta=0A= =0A= =0A= =0A= =0A= > Subject: categories: Re: Terminology: Remarks=0A= > From: jean.benabou@wanadoo.fr=0A= > Date: Fri=2C 3 May 2013 06:53:12 +0200=0A= > CC: categories@mta.ca=0A= > To: categories@TobyBartels.name=0A= > =0A= > Dear Toby=2C=0A= > I'm preparing an answer to all the mails I received about equivalence of = categories. In order to answer to yours=2C I need the following precisions = about your definition:=0A= > =0A= > (i) If F: A -> B and G: B -> C are full and faithful essentially surjecti= ve functors=2C so is GF. How do you compose your equivalences?=0A= > (ii) Let 1 denote the final category. The unique functor 1 -> 1 is the un= ique equivalence between 1 and 1. How many spans 1 <- X -> 1 are equivalenc= es in your sense?=0A= > =0A= > Best regards=2C=0A= > Jean=0A= > =0A= > =0A= > Le 2 mai 2013 =E0 08:46=2C Toby Bartels a =E9crit :=0A= > =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= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]