M Theory Lesson 104

Another way to see why a map $Q \rightarrow P$ between cubes and permutohedra is natural is to consider the parity cube labelled by bracketings of four objects, which is the way it naturally appears in the tricategorical axioms. The lower left vertex represents an object $A \otimes B \otimes C \otimes D$ with no specified bracketing. A first choice of bracketing leads to an unambiguous diagram, but further choices result in multiple diagrams. The target vertex is the full hexagon of levelled trees for the symmetric group $S_3$. Pentagons appear by forgetting the level information of the front top right vertex. The front face is also the deformation of the pentagon as represented by five sides of the parity cube.

Observe that this relates a three dimensional cube to a two dimensional hexagon, and is thus more like a homological connecting morphism (decategorification) than a Loday map, which operates in a fixed dimension. Conversely, the hexagon poset may be decomposed as shown to form the cube in one dimension higher.