M Theory Lesson 152

In a 2002 paper, in referring to another paper which never appeared, Batanin mentions a topological 2-operad containing the sequence of permutohedra, which are labelled by extended 1-ordinal trees as shown. The sequence begins with the interval, the hexagon and the hexagonal and square faced 24 vertex polytope. Notice that the 2-ordinal labelling here is different from the more modern one, which Batanin used to solve the general compactification problem. For example, the hexagon should be replaced by the Breen polytope, a double 12 sided object, which may be resolved into a 12 sided cylinder in the B operad.

But sticking with the old example, the last polytope (labelled by the 4 leaved tree) maps to the 3d Stasheff associahedron (labelled by a single level 4 leaved tree) under a Loday type map, which forgets the levels on the trees that are used to label permutations. So the Loday-Ronco triples are based on 1-ordinal sequences, whereas we would like to view the permutohedra as part of a 2-operad, and similarly the cubes as part of a 3-operad. The old example actually considers a 2-operad in Cat, and another operad in Span(Cat) (spans in the category of categories), the algebras of which give the sought after Gray categories. If anyone has further references to such examples, I would really appreciate finding them!

Now a 3-ordinal tree with only three leaves, which looks like a central extension of the 2-ordinal tree which usually labels the hexagon, happens to label a hexagon of the form shown, which came up recently in lessons when we tried to tile Riemann surfaces with associahedra. So maybe this silly hexagon on a pair of pants really is trying to tell us something. We know we want it to come from a 3-operad eventually, because mass generation is characterised by Gray type structures.