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/ ]