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

[JmsNxn]

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.

Lmao!

Hey, Mphlee!

This is standard Analytic Number Theory. This is how Analytic Number Theory works. So if you are finding some relation to Category theory--that is the relation that Analytic Number Theory has to Category Theory. Honestly, I think that's why I understand you better than most people here. I have a strong background in Number Theory/Analytic Number Theory--and these are discrete modes of research. And there's a fairly strong overlap in many ways. I'm pretty sure I just rediscovered things I have read before, I'm just too dumb to remember where I read it. I'm willing to bet the past 3 posts were 90% already proved by greater mathematicians than me  

I'll peruse your links though, always love your comments

EDIT: Just want to say I know what a characteristic function is. And I know what a radical is. But I didn't think this was a radical, so that's super cool, that this is a radical... lol I only know radicals from Analytic Number Theory, and I def wouldn't of called my function that. Still super cool! Thanks for your comments!!!!!!!