Re: bilax_monoidal_functors

[Andre Joyal <joyal.andre@uqam.ca>]

Dear Urs and John,

I see no real conflict between your terminology and mine.

I do use the notion of n-fold monoid in my work, for example
in my "Notes on Quasi-categories".
An *algebraic theory* is defined to be a (quasi-)category with finite products.
The n-fold tensor power of the theory of monoids M
is the theory of n-fold monoids = E(n)-monoids for every n.

I am sketching a proof of the Stabilisation Hypothesis at section 43.5 of my notes.
The hypothesis is formulated in terms of an equivalence of theories: 

<The theory of (n+2)-fold monoidal n-categories is equivalent 
to the theory of symmetric monoidal n-categories>.

It follows that the quasi-category of (n+2)-fold monoidal n-categories 
is equivalent to the quasi-category of symmetric monoidal n-categories.

Best, 
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Urs Schreiber
Date: lun. 10/05/2010 06:28
À: John Baez
Objet : categories: Re: bilax monoidal functors
 
> An n-tuply monoidal k-category is (conjecturally) a special sort of
> (n+k)-category

By the way, some progress on this is available from John Francis' and
Jacob Lurie's discussion of k-tuply monoidal (n,1)-categories as those
equipped with a little k-cubes action.

In particular there is a proof of the stabilization hypothesis for
(n,1)-categories this way, and an analog of the May recognition
theorem for parameterized oo-groupoids, i.e. (oo,1)-sheaves.

Some of this is summarized with pointers to references here:

   http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category

Best,
Urs

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