The stabilisation theorem

["Joyal, André" <joyal.andre@uqam.ca>]

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/ ]