Discussion on "tetra-eta-series" (2007) in MO

[MphLee]

So let \(\chi_m(n) = 1\) if there exists a \(k\) where \(m = n^k\), and zero other wise.

Given the notation and this definition this is literally a characteristic function.
I leave this in the thread only as a terminological addendum. Let \(A\subseteq X\) then it's characteristic function \(\chi_A:X\to 2\) is defined as follows: \(\chi_A(x) = 1\) if \(x\in A\), and zero other wise. The concept is deeply categorical... and also an ancient one.
In your case just define for every \(m\in\mathbb N\) the set \(\sqrt m:=\{n\in \mathbb N\, |\,\exists k.\, n^k=m\}\), then its characteristic function is the function you've defined, i.e. \(\chi_m(n) := \chi_{\sqrt m}(n)\), this and old old concept.

Now you are moving at the level of numbers... categorifying means switching pojnt of view and instead of studying numbers directly... that are the shadows of some higher level combinatorics business, you directly aim your attention at the set theoretic properties of the mapping \(m\to \sqrt m\) as \(m\) varies. This is basically what seems you are doing.. but implicitly. It turns out, I've studied this under the name of intrinsic iteration (here and here I hinted at it), that this construction is functorial... and is deeply related to rational iteration (how? by considering exactly the same sets but instead of over the multiplicative monoid of natural numbers you do it over arbitrary monoids, e.g. monoids of functions under composition).

PS: I apologize for not being able to make say something more interesting but I'm lacking mental and physical vigor to do any better.... I'm sorry because I think this is really fascinating.