From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6895 Path: news.gmane.org!not-for-mail From: =?UTF-8?Q?Jonathan_CHICHE_=E9=BD=8A=E6=AD=A3=E8=88=AA?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Simplicial versus (cubical with connections) Date: Wed, 14 Sep 2011 09:08:13 +0200 Message-ID: References: Reply-To: =?UTF-8?Q?Jonathan_CHICHE_=E9=BD=8A=E6=AD=A3=E8=88=AA?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1316029123 25579 80.91.229.12 (14 Sep 2011 19:38:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 14 Sep 2011 19:38:43 +0000 (UTC) To: Categories list , Ronnie Brown Original-X-From: majordomo@mlist.mta.ca Wed Sep 14 21:38:37 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 1R3vI4-0004D5-R5 for gsmc-categories@m.gmane.org; Wed, 14 Sep 2011 21:38:36 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42064) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1R3vGZ-0008G3-2h; Wed, 14 Sep 2011 16:37:03 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1R3vGW-0003Vq-R9 for categories-list@mlist.mta.ca; Wed, 14 Sep 2011 16:37:00 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6895 Archived-At: There is another way to state that the cube category with connections =20= behaves "as well as" the simplex category. Both are strict test =20 categories (as defined by Grothendieck in "Pursuing Stacks"). See =20 http://www.math.jussieu.fr/~maltsin/ps/cubique.pdf. Without =20 connections, the cube category is a test category, but not a strict =20 one, so that the product in the cube category does not reflect the =20 product of homotopy types. This issue vanishes if connections are =20 allowed. Grothendieck explicitly wrote in "Pursuing Stacks" that he =20 believed that, homotopically speaking, any strict test category was =20 "as good as" the simplex category. For instance, he conjectured there =20= that an analog of the Dold-Kan correspondence (which he called Dold-=20 Puppe) holds for every strict test category. (As regards the =20 existence of a Quillen model structure the cofibrations of which are =20 monomorphisms on the presheaf category, and so on, see the =20 introduction to Ast=E9risque 301 by Maltsiniotis and Ast=E9risque 308 by = =20 Cisinski.) Best regards, Jonathan Chiche= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]