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/ ]