Michael Batanin has kindly given me permission to put up some of the beautiful pictures from his talk on Wednesday at AustMS 06. This resultoassociahedron has probably not appeared on the web before. Notice the indexing by a two-level tree, as mentioned yesterday.

Here is some more combinatorics from Dinner at the End of the Universe…

There are quite a number of operads floating around, such as this B operad…

The B operad is a true n-dimensional generalisation of the Stasheff one for 1-fold loop spaces. The basic K diagrams themselves do not form an operad, for a reason that we will discuss in the future. At the end of his talk, Batanin discussed the Baez-Dolan stabilisation hypothesis, for which there is now evidence.

Isn’t this great! Now we can start calculating things in M-theory using simple manipulations of polytopes.

## david said,

September 29, 2006 @ 11:29 am

Has Batanin explained how that chart marked ‘Getzler-Jones, Baez-Dolan, Batanin’ works? Presumably we should complete the top left corner with -1 and -2 columns, as John has explained on p. 12 of this. I guess the -1 column has to begin |, and the -2 column with nothing.

## Kea said,

September 29, 2006 @ 11:57 am

Hi David

Good thought. I haven’t worked through it myself yet. As for the table: count the edges (respecting levels) of each tree. The degree goes like #e – d – 1.

## Kea said,

September 30, 2006 @ 5:56 am

…goes like#e – d – 1This comes from something called Fox-Neuwirth cells for n-ordinals.

## david said,

September 30, 2006 @ 9:58 am

In view of stability arriving in the third row of the 1-column, perhaps that’s more like the Baez-Dolan 0-column, i.e., the one that goes Set – Monoid – Commutative Monoid – … Then the 0-column would be their -1 – column stabilizing after 2 rows, and we would only have to put in a -1-column, beginning with |, and so already stable.

## Kea said,

September 30, 2006 @ 10:51 am

Yes, Batanin leaves off Set, Cat etc. at h=0 so in column 2 one has first the pentagon then the hexagons and then symmetry (see the trees), which means column 1 is as you say, I guess. Cool.

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