M Theory Lesson 56
$\omega (n,a) + \omega (n,b) = \omega (n,a+b)$
To a category theorist this classical relation looks like a triangle, where two sides compose to give the third. Moreover, since $n$ is fixed, we may view $\omega(n,-)$ as a functor from a category containing the triangle $(a,b,a+b)$. This source category has addition as composition. There is only one object. In fact, it is an Abelian group in the form of a category on one object. The target category, containing the frequencies as arrows, is also an Abelian group, so the functor $\omega (n,-)$ is just a group homomorphism.
Following Heisenberg, the quantum relation takes the form
$\omega (n, n-a) + \omega (n-a, n-a-b) = \omega (n, n-a-b)$
Setting $a = 0$ we see that $\omega (n,n)$ acts as a (left) identity for frequency composition, replacing the classical $\omega (n,0)$. It is now necessary to consider a variable $n$ in the source category of the classical case, so frequencies are best expressed as 2-arrows between 1-arrow integers. Thus the honeycomb diagrams in the plane are associated with two dimensional categorical structures. Alternatively, and loosely speaking, a set theoretic real number classical physics suggests a categorical complex number quantum physics.