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.
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.
posted by Kea | 5:49 PM
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