From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6754 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 09:21:19 -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 1310312265 4764 80.91.229.12 (10 Jul 2011 15:37:45 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 10 Jul 2011 15:37:45 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Sun Jul 10 17:37:36 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 1Qfw4c-0005TG-Kr for gsmc-categories@m.gmane.org; Sun, 10 Jul 2011 17:37:34 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:60946) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Qfw3B-0005vo-3C; Sun, 10 Jul 2011 12:36:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qfw3A-00085J-Bu for categories-list@mlist.mta.ca; Sun, 10 Jul 2011 12:36:04 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6754 Archived-At: Dear Marta, I thank you for your message and for drawing my attention to your work. I apologise for not having refered to it. >More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried out =20= the 2-analogue of the 1-dimensional >case along the same lines of the 1979 papers, by constructing the 2-=20= 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 topos S. That is a good question. I am not aware that this has been done (but =20 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 -------- Message d'origine-------- De: Marta Bunge [mailto:martabunge@hotmail.com] Date: ven. 08/07/2011 09:00 =C0: Joyal, Andr=E9; edubuc@dm.uba.ar; categories@mta.ca Objet : RE : categories: size_question_encore Dear Andre,I welcome your suggestion of involving stacks in order to =20 test universality when the base topos S does not have Choice. I have =20= been exploiting this implicitly but systematically several times since =20= my own construction of the stack completion of a category object C in =20= any Grothendieck topos S (Cahiers, 1979). For instance, I have used it =20= crucially in my paper on Galois groupoids and covering morphisms =20 (Fields, 2004), not only in distinguishing between Galois groupoids =20 from fundamental groupoids, but also for a neat way of (well) defining =20= the fundamental groupoid topos of a Grothendieck topos as the limit of =20= a filtered 1-system of discrete groupoids, obtained from the naturally =20= arising bifiltered 2-system of such by taking stack completions. This =20= relates to the last remark you make in your posting. Concerning S_fin, =20= it does not matter if, in constructing the object classifier, one uses =20= its stack completion instead, since S is a stack (Bunge-Pare, Cahiers, =20= 1979). In my opinion, stacks should be the staple food of category =20 theory without Choice. For instance, an anafunctor (Makkai's =20 terminology) from C to D is precisely a functor from C to the stack =20 completion of D. More recently (Bunge-Hermida, MakkaiFest, 2011), we =20 have carried out the 2-analogue of the 1-dimensional case along the =20 same lines of the 1979 papers, by constructing the 2-stack completion =20= of a 2-gerbe in "exactly the same way". Concerning this, I have a =20 question for you. Is there a model structure on 2-Cat(S) (or 2-=20 Gerbes(S)), for S a Grothedieck topos, whose weak equivalences are the =20= weak 2-equivalence 2-functors, and whose fibrant objects are precisely =20= the (strong) 2-stacks? Although not needed for our work, the question =20= came up naturally after your paper with Myles Tierney. We could find =20 no such construction in the literature. With best regards, Marta > Subject: categories: RE : categories: size_question_encore > Date: Wed, 6 Jul 2011 21:23:36 -0400 > From: joyal.andre@uqam.ca > To: edubuc@dm.uba.ar; categories@mta.ca > > Dear Eduardo, > > I would like to join the discussion on the category of finite sets. >= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]