M Theory Lesson 31

Ternary logic, as introduced on kneemo's blog, is based on functions between three objects: 0, 1 and 2. Stepping back to the two object case, we see that there are four possible ways of mapping 0 and 1 into the two element set, which plays the role of subobject classifier in the topos Set. The values 00, 01, 10 and 11 form the four vertices of the parity square. It was the combinatorics of this simple square that led Gray to develop 2-categories and the Gray tensor product in the 1960s.

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...