We see that noncommutative paths add one dimension to number spaces, just as the quaternions are one complex dimension larger than the complex numbers. Similarly, the diagram shows how nonassociativity adds two (complex) dimensions to the quaternions. It maps the path $XXX$ to the path $XXY$, which is just an edge in the lattice of noncommutative paths.

The square of bracketings is a categorical associator square, and the four object version will break the monoidal structure of a higher category. This square is a natural transformation between functors, so we should not discuss nonassociativity outside of a higher categorical context.

8 years ago

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