From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6784 Path: news.gmane.org!not-for-mail From: =?ISO-8859-1?Q?Andr=E9_Joyal?= Newsgroups: gmane.science.mathematics.categories Subject: A new axiom? Date: Wed, 13 Jul 2011 10:32:30 -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 1310580169 4276 80.91.229.12 (13 Jul 2011 18:02:49 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 13 Jul 2011 18:02:49 +0000 (UTC) Cc: mshulman@ucsd.edu, david.roberts@adelaide.edu.au, categories To: martabunge@hotmail.com Original-X-From: majordomo@mlist.mta.ca Wed Jul 13 20:02:44 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Qh3lk-0001E9-5s for gsmc-categories@m.gmane.org; Wed, 13 Jul 2011 20:02:44 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:41817) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1Qh3kF-0003wq-EX; Wed, 13 Jul 2011 15:01:11 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qh3kE-00061R-Qr for categories-list@mlist.mta.ca; Wed, 13 Jul 2011 15:01:10 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6784 Archived-At: Dear Marta and all, The category of simplicial objects in a Grothendieck topos admits a =20 model structure in which the weak equivalences are the local weak homotopy =20 equivalences and the cofibrations are the monomorphisms (I have described the model =20= structure in my 1984 letter to Grothendieck). A higher stack can be defined to be a simplicial object which is =20 globally homotopy equivalent to a fibrant object. The notion of internal simplicial object can be defined in any =20 elementary topos with natural number object. The local weak homotopy equivalences between simplicial objects can be =20= defined internally. It seems reasonable to introduce a new axiom for an elementary topos E =20= (with natural number object). It may be called the Model Structure Axiom: The MSA axiom: "The category of simplicial objects in E admits a model =20= structure in which the weak equivalences are the local weak homotopy =20 equivalences and the cofibrations are the monomorphisms" A nice thing about this axiom is that it implies the existence of n-=20 stack completion for every n. It also implies the existence of infinity-stack completion. Best, Andr=E9= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]