M Theory Lesson 31
A unary arrow $0 \rightarrow 1$, thought of as a category, is used to lift morphisms in any 1-category to functors. Similarly, we would expect the square to form pseudofunctors for morphisms. This is what happens when the Mac Lane pentagon is lifted up to the sides of the parity cube.
Given that Gray based his whole theory of 2-categories on the parity square, it is natural to ask what would happen with the parity cube. After all, QCD prefers ternary logic. As kneemo noted however, things are immediately different for the cube. There is only one possible square on the two letters 0 and 1, but there are three possible cubes on the letters 0, 1 and 2. This brings a notion of triality into this higher categorical structure, which we expect will clarify the division algebras.
So it seems that all of 20th century physics comes from understanding nothing more than the low prime number 3. Then there are more primes...