Is there some categorical version of tetration or higher hyperoperations?
This is motivated by the fact that coproducts categorify addition of finite cardinals, and products/exponentials categorify multiplication/exponentiation in the same way. (exponentials categorify infinite cardinal exponentiation too)
Hunting for intuition in $Sets$ works for the first three operations, so it seems reasonable to look there for intuition about higher operations; Will Brian gives this intuition for cardinals here, which I’ll copy:
$\kappa \!\uparrow\uparrow\! \nu$ is equal to the number of $B$-names of rank at most $\nu$, where $B$ is a set of size $\kappa$.
Can we convert this canonically into a universal property for a ‘tetrate’ of two objects in a category/two categories? How about a cardinal definition for higher operations?
I had hoped naïvely that a ‘tetrate by a fixed object’ functor $^{A}-$ would fit in the adjoint chain $-\times A \dashv -^{A}\dashv{^{A}-}$, but the right adjoint of $-^{A}$ is well studied in Synthetic Differential Geometry under the name $(-)_A$, for example here. It seems unlikely that a tetration functor would be intrinsic to reasoning about smooth phenomena so this definition looks incorrect for the desired intuition, but I still wonder if defining tetration via an adjunction might be easier.
