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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]