Operadification II

Underlying the concept of natural number object is the basic recursion theorem. The composition of the arrows

$f\circ q:1\to A\to A$

in Set is just the evaluation $f\left(q\right)$. This arrow $f\left(q\right):1\to A$ can itself be used as input for the same diagram, by appending another copy of $f$ to the right, to obtain the arrow $f\left(f\right(q\left)\right)$. That is, the natural number object commuting diagram extends indefinitely to the right by appending extra copies of the successor and the function $f$. Once the comparison arrow $u:N\to A$ assigns zero to $q$, it follows that it must assign $f\left(q\right)$ to 1, $f\left(f\right(q\left)\right)$ to 2, and so on.

Thus the definition of recursion, as a possibly infinite process, demands the full set $N$ rather than some finite ordinal set. But for periodic recursive functions, satisfying $f\left(f\right(f(\cdots (q\left)\right)\left)\right)=f\left(q\right)$ for $n+1$ brackets on the left hand side, modular arithmetic using the set n acts as a universal diagram. For example, if $f$ represents rotation by an $n$-th root of unity, then it is periodic in this sense.