I really admire Baez’s groupoidification program, but in certain contexts, a groupoid is too restrictive. This is mainly because composition of morphisms is still associative.

Recently I’ve been thinking it would be helpful to categorify the Cayley-Dickson process. For at each step of this process, we see we lose properties such as: every element is its own conjugate, commutativity, associativity, and the division property. If we were taking an n-Cat approach, by the third Cayley-Dickson doubling we couldn’t even make a 3-groupoid! This would force us to resort to using quasigroupoids or other weaker structures to build our higher “*-algebroids” (Sedenions, etc).

What’s cool is that when we go higher than the 3-Cat level, our n-cells include zero-divisors, i.e., we can have fg=0 for non-zero n-cells f and g. I’ve seen multiplication tables for the *-algebras higher than the octonions (sedenions, 32-ions, 64-ions, 128-ions, etc.) and they display fractal self-similarity after a while. It would probably be fun to explore these using n-Cats.

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