### M Theory Lesson 183

Recall that the sixth face of the parity cube may represent a breaking of the Mac Lane pentagon by splitting the symmetric four leaved tree into two parts. This tree was also considered by Forcey et al in a 2004 paper discussing higher operads, beginning with the observation indicated by the following diagram. Consider the boxed vertical lines as a fixed object in the category, and ignore the bottom third of the diagram. Then there are two ways to piece together the tree: do the horizontal (pink) products first, or else the vertical (green) ones. This issue of commutativity for two tensor products is a central axiom of a bicategory, commonly called the interchange rule. By considering categories with three products, Forcey et al magically go on to prove that (ordered) three dimensional Young diagrams can describe what they call a 3-fold monoidal category, a fascinating recursive structure. Moreover, this result generalises to all higher dimensions.

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