From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6756 Path: news.gmane.org!not-for-mail From: =?ISO-8859-1?Q?Andr=E9_Joyal?= Newsgroups: gmane.science.mathematics.categories Subject: RE: size_question_encore Date: Sun, 10 Jul 2011 09:30:07 -0400 Message-ID: Reply-To: =?ISO-8859-1?Q?Andr=E9_Joyal?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain;charset=ISO-8859-1;format=flowed;delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1310312518 6114 80.91.229.12 (10 Jul 2011 15:41:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 10 Jul 2011 15:41:58 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Sun Jul 10 17:41:54 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Qfw8m-0007Yh-UL for gsmc-categories@m.gmane.org; Sun, 10 Jul 2011 17:41:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:60360) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Qfw7B-0006IV-Ku; Sun, 10 Jul 2011 12:40:13 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qfw7A-0008Cc-WC for categories-list@mlist.mta.ca; Sun, 10 Jul 2011 12:40:12 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6756 Archived-At: dear Marta, I apologise, I had forgoten our conversation! My memory was never good, and it is getting worst. You wrote: >No, I am not thinking of the analogue of Steve Lack's model =20 structure since, strictly speaking, >it has nothing to do with stacks. Comments to that effect (with =20 which Steve agrees) are included >in the Bunge-Hermida paper. It was actually a surprise to discover =20 that after trying to do what >you suggest and failing. I disagree with your conclusion. I looked at your paper with Hermida. We are not talking about the same model structure. The fibrations in =20 2Cat(S) defined by Steve Lack (your definition 7.1) are too weak when the topos S does =20= not satify the axiom of choice. Equivalently, his generating set of trivial cofibrations is too small. Nobody has read my paper with Myles . Best, Andr=E9 -------- Message d'origine-------- De: Marta Bunge [mailto:martabunge@hotmail.com] Date: sam. 09/07/2011 13:41 =C0: Joyal, Andr=E9; edubuc@dm.uba.ar; categories@mta.ca Objet : RE: RE : RE : categories: size_question_encore Dear Andre, You were indeed aware of my work and that with Pare on stacks since =20 you are one of the few we thank for useful conversations! There were =20 two ways to define stacks and one of them was your suggestion. One =20 could say that one is expressed directly in terms of descent and the =20 other in terms of weak equivalences. As it turns out, both are needed =20= in my construction of the stack completion and similarly in the 2-=20 dimensional case. As for which method is preferable, I do not know. Whether one =20 constructs stack completions for categories in a Grothendieck topos =20 using the carving out from presheaf toposes (my method), or by means =20 of a model structure (yours), one has to resort to the existence of a =20= generating family to keep them small. No, I am not thinking of the analogue of Steve Lack's model structure =20= since, strictly speaking, it has nothing to do with stacks. Comments =20 to that effect (with which Steve agrees) are included in the Bunge-=20 Hermida paper. It was actually a surprise to discover that after =20 trying to do what you suggest and failing. I attach my paper with =20 Hermida in this connection. Section 3 makes clear what happens with =20 Lack's model structure in dimension 1, and Section 7 considers the 2-=20 dimensional analogue, also not suitable to get the 2-stack completion. I really meant an extension of the Joyal-Tierney model structure. =20 Thanks for pointing out Moerdijk's work, and your old one with =20 Tierney. I will eventually look into those. No need to respond to this. Best regards, and many thanks,Marta > Subject: RE : RE : categories: size_question_encore > Date: Sat, 9 Jul 2011 12:18:45 -0400 > From: joyal.andre@uqam.ca > To: martabunge@hotmail.com; edubuc@dm.uba.ar; categories@mta.ca > > Dear Marta, > > I thank you for your message and for drawing my attention to your =20 work. > I apologise for not having refered to it. > > >More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried =20 out the 2-analogue of the 1-dimensional > >case along the same lines of the 1979 papers, by constructing the =20= 2-stack completion of a 2-gerbe in "exactly the same way". >Concerning =20= this, I have a question for you. Is there a model structure on 2-=20 Cat(S) (or 2-Gerbes(S)), for S a Grothedieck topos, >whose weak =20 equivalences are the weak 2-equivalence 2-functors, and whose fibrant =20= objects are precisely the (strong) 2-stacks? >Although not needed for =20= our work, the question came up naturally after your paper with Myles =20 Tierney. We could find no such >construction in the literature. > > I guess you are thinking of having the analog of Steve Lack's model =20= structure > but for the category of 2-categories internal to a Grothendieck =20 topos S. > That is a good question. I am not aware that this has been done =20 (but my knowledge of the litterature is lacunary). > You may also want to establish the analog of Moerdijk's model =20 structure for the category of internal 2-groupoids. > I am confident that these model structure exists. > They should be closely related to a model structure on internal =20 simplicial groupoids > . > And also related to the model structure on simplicial sheaves, =20 described in my letter > to Grothendieck in 1984, but unfortunately not formally published. > > Best regards, > Andre > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]