Re: calculus, homotopy theory and more (corrected)

[Michael Batanin <mbatanin@ics.mq.edu.au>]

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.




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