From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6785 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: RE: A new axiom? Date: Wed, 13 Jul 2011 11:41:43 -0400 Message-ID: References: <0D6B50AE-6C16-48CD-A75C-A4280FCE7FF0@uqam.ca> Reply-To: Marta Bunge NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1310580264 4951 80.91.229.12 (13 Jul 2011 18:04:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 13 Jul 2011 18:04:24 +0000 (UTC) Cc: Mike Shulman , David Roberts , To: Original-X-From: majordomo@mlist.mta.ca Wed Jul 13 20:04:17 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 1Qh3nF-00027d-AM for gsmc-categories@m.gmane.org; Wed, 13 Jul 2011 20:04:17 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:41823) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1Qh3lC-00048P-Qa; Wed, 13 Jul 2011 15:02:10 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qh3lC-00064O-4N for categories-list@mlist.mta.ca; Wed, 13 Jul 2011 15:02:10 -0300 In-Reply-To: <0D6B50AE-6C16-48CD-A75C-A4280FCE7FF0@uqam.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6785 Archived-At: Dear Andre=2C Sounds good=2C except that I do not quite see what the status of such an ax= iom is if added to ET + NNO. I do not doubt that any GT satisfies it.=A0 An alternative=2C which I had in mind and on which spoke about it in my 1 h= our lecture at Calais 2008=2C is to add (ASC)^n for each n>0 to ET + NNO. H= ow to state (ASC)^n in elementary terms relies on Lemma 8.2 of Bunge-Hermid= a=2C of which I have sketched a proof by induction. It says that for every = epimorphism e from J to I in E=2C the induced n-functor (F_e)^n from the n-= kernel of e to the discrete n-category on I is a weak n-equivalence n-funct= or (as defined in Def. 8.1). Intended definition: An n-category C in E is a= n n-stack if for every epimorphism e=2C C inverts (F_e) in the sense of n-e= quivalence. This is an elementary axiom for each n >0=2C =A0and one could a= dd as many as one needed for a specific purpose. In constructing the 2-stac= k completion=2C I only needed the n=3D1 case. So=2C for 3-stack completions= =2C n =3D 1 and n=3D2 would suffice. Etc. Beyond that I cannot envisage any= uses of n-stacks. But then=2C I am not a higher-order category person. In any case=2C this is getting interesting.=A0 Best regards=2C Marta > From: joyal.andre@uqam.ca > To: martabunge@hotmail.com > Subject: A new axiom? > Date: Wed=2C 13 Jul 2011 10:32:30 -0400 > CC: mshulman@ucsd.edu=3B david.roberts@adelaide.edu.au=3B categories@mta.= ca >=20 > Dear Marta and all=2C >=20 > 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. >=20 > 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. >=20 > 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: >=20 > 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" >=20 > 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. >=20 > Best=2C > Andr=E9 = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]