The stabilisation theorem

The stabilisation theorem

[Joyal, André]

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



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Re: The stabilisation theorem

[John Baez]

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


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Re: The stabilisation theorem

[joyal.andre]

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/ ]