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.
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.
ReplyDeleteHi David
ReplyDeleteGood 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.
...goes like #e - d - 1
ReplyDeleteThis comes from something called Fox-Neuwirth cells for n-ordinals.
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.
ReplyDeleteYes, 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.
ReplyDelete