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, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
André wrote: Too simple to be true?
No, I always hoped for a simple proof! And this proof is not "too simple". It seems all the hard work is packed into the formalism that underlies the proof. And that's how it should be, I think. So, I'm happy. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear John,
It seems all the hard work is packed into the formalism that underlies the proof.
I cannot resist describing a key idea of my proof. It is to construct categories from homotopy types rather than from sets. The notion of category is homotopy essentially algebraic: "A category is essentially the same thing as a complete Segal object X satisfying an extra condition: the (source, target) map X_1-->X_0 times X_0 is a 0-cover (a map is a 0-cover if its homotopy fibers are discrete). This is like constructing the "natural" (or folk") model structure on Cat from a model structure on simplicial diagrams of spaces (spaces = simplical sets). A similar description applies to (weak) n-categories. The Stabilisation Hypothesis was a great conjecture. Let me congratulate you and Jim Dolan for formulating it. Best, André -------- Message d'origine-------- De: categories@mta.ca de la part de John Baez Date: dim. 16/05/2010 14:44 À: categories Objet : categories: Re: The stabilisation theorem André wrote: Too simple to be true?
No, I always hoped for a simple proof! And this proof is not "too simple". It seems all the hard work is packed into the formalism that underlies the proof. And that's how it should be, I think. So, I'm happy. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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John Baez -
Joyal, André -
joyal.andre＠uqam.ca