From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5795 Path: news.gmane.org!not-for-mail From: Michael Batanin Newsgroups: gmane.science.mathematics.categories Subject: Re: calculus, homotopy theory and more (corrected) Date: Fri, 14 May 2010 08:59:55 +1000 Message-ID: References: <4BE81F26.4020903@dm.uba.ar> Reply-To: Michael Batanin NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1273843296 2978 80.91.229.12 (14 May 2010 13:21:36 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 14 May 2010 13:21:36 +0000 (UTC) To: Joyal@ics.mq.edu.au, =?ISO-8859-1?Q?Andr=E9?= Original-X-From: categories@mta.ca Fri May 14 15:21:34 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 1OCupY-0003G1-UB for gsmc-categories@m.gmane.org; Fri, 14 May 2010 15:21:33 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OCuNs-0002VS-4U for categories-list@mta.ca; Fri, 14 May 2010 09:52:56 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5795 Archived-At: Dear Andre, > *If a n-type has the structure of an E(n+2)-space then it has > the structure of an E(infty)-space (canonically)* > > > which follows from the fact that the E(n+2) operad is n-connected. > You may recall that we have discussed this in Barcelona. > You told me that you knew that the E(n+2) operad is n-connected. > I had learned it a week before from Lurie during my visit to Toen in Toulouse. Yes, I remember this discussion. Actually my proof comes down to the same fact since Q_n has homotopy type of unodered little cube configurations in an n-cube. Lurie's proof is also based on the same fact. Another approach to the proof that n-type with E_{n+2}-space structure is also E_{infty}-space can be obtained by combining an idea of John Baez and Jim Dolan of counting n-trees and my calculations of cells in the Fulton-Macpherson operad. This is extremely simple combinatorial proof. I can not reproduce it in this post because it requires some pictures. But I remember, Andre, we discussed it with you in Montreal in 2004. I'll be happy to explain it again in Genoa. > The rest of the proof of the Stabilisation hypothesis is formal but > depends heavily on the machinery of (homotopical) universal algebra > I have developed in my "Notes on quasi-categories". The same for our proof. It does require a lot of homotopical algebra to be able to localize model categories of operads. > I believe that Goodwillie calculus is one of the next big thing in math. I agree with it. It would be really wonderful if some experts organize a workshop on this subject with some introductory lectures for the beginners. > > I look forward to see you in Genoa, So do I. Michael. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]