Re: calculus, homotopy theory and more (corrected)
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/ ]
Dear Michael, A basic ingredient in my approach to higher categories is the notion of complete Segal space introduced by Rezk. I have learned Rezk theory in proving the Quillen equivalence between quasi-categories and complete Segal spaces. In my "Notes on Quasi-categories" I am introducing an abstract notion of complete Segal space called *Rezk category*, or *reduced category*. A category object (internal to a quasi-category) is said to be *reduced* if its object of objects is *isomorphic* to its object of isomorphisms via the unit map. (an isomorphism in a quasi-category is an arrow which is invertible in the homotopy category). An ordinary category (in set) is reduced iff every isomorphism is a unit, a very stringent condition. Ordinary categories are seldom reduced (posets are). Every reduced category is skeletal. An equivalence between reduced categories is necessarly an isomorphism. In contrast, there are plenty of reduced categories in homotopy theory. In fact every category internal to the quasi-category of spaces is *equivalent* to a reduced category (via a fully faith ess surj functor). This key result was proved by Rezk for complete Segal spaces: every Segal category is *equivalent* to a complete Segal space. The theory of reduced categories is essentially (homotopy) algebraic (unlike ordinary category theory in which we need to expand the notion of isomorphism (of categories) with that of equivalence). I do not have the time to explain more of the idea of my proof now. A sketch can be found in my "Notes on Quasi-categories". You wrote:
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.
I hope I will understand this time! I always find our conversation very stimulating! See you in Genoa, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Andre Joyal -
Michael Batanin