From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5809 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: The stabilisation theorem Date: Sat, 15 May 2010 11:06:20 -0300 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= 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 1274023928 18816 80.91.229.12 (16 May 2010 15:32:08 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 16 May 2010 15:32:08 +0000 (UTC) To: Bob Rosebrugh Original-X-From: categories@mta.ca Sun May 16 17:32:06 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ODfp0-0008CL-DA for gsmc-categories@m.gmane.org; Sun, 16 May 2010 17:32:06 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ODfJg-0006Va-D9 for categories-list@mta.ca; Sun, 16 May 2010 11:59:44 -0300 Thread-Topic: The stabilisation theorem(corrected) Thread-Index: Acrxcbq3yJCUXKa2SKa0kYIlwngJtQCxKEyM Content-Language: en-US Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5809 Archived-At: Dear John, I wrote: >> The Breen-Baez-Dolan Stabilisation Hypothesis is a theorem. You wrote: >It seems I understand everything except this sentence. I have a pretty good idea of how it can be proved. It is like a road map. The steps are not difficult to understand. Here is a sketch. 1) Let me denote by E(n) the theory of n-fold monoids and by E(infty) the theory of symmetric monoids. If U[n] dnotes the quasi-category of n-types, then the map Model(E(infty),U[n]) --->Model(E(n+2), U[n]) induced by the canonical map E(n+2)-->E(infty) is an equivalence of quasi-categories. This follows from the fact that the map E(n+2)-->E(infty) is a n-equivalence. 2) If T is any finite limit sketch, then the equivalence above induces an equivalence of quasi-categories Model(E(infty),Model(T,U[n])) --->Model(E(n+2), Model(T,U[n])) In particular, if T is the theory of n-categories T_n, we obtain an equivalence of quasi-categories Model(E(infty),Cat(n)) --->Model(E(n+2), Cat(n)) where Cat(n) is the quasi-category of (weak)-n-category. QED Too simple to be true? I am ready to give more details if you want. Best regards, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]