Is there a category-theoretic framework for comparing asymptotic growth rates (e.g. results like the Prime Number Theorem)?

Many classical order-theoretic and topological facts admit clean categorical reformulations: suprema as colimits, density theorems via the Yoneda embedding, metric spaces as categories enriched over $([0,\infty], \geq, +)$ (Lawvere), etc. These capture structural facts well.

However, results whose content is fundamentally about comparing the asymptotic growth of two unbounded quantities — e.g. $\pi(x) \sim x/\ln x$ — don't seem to fit naturally into this mold, even after enriching over a quantale. Enrichment gives you a notion of "distance" or "cost" between two objects, but not an obvious notion of comparing growth rates of two objects as some parameter tends to infinity.

I'm aware that coarse geometry (Roe, Higson–Roe) categorifies large-scale/asymptotic behavior of metric spaces via coarse maps and coarse equivalence, and that magnitude of enriched categories (Leinster) gives a categorically-native numerical invariant. Neither seems to directly address comparing growth rates of number-theoretic functions.

So i have three questions:

1. Is there an existing categorical/enriched framework designed specifically to express or prove asymptotic comparison statements (like $f(x) \sim g(x)$ or $f = O(g)$) natively, rather than importing them as external numerical facts?
2. Does coarse geometry, or some other existing framework, actually capture statements like the Prime Number Theorem, or is it fundamentally out of reach of current categorical machinery?
3. Is this a known open direction, or is there a structural reason (e.g. related to the fact that "field" isn't algebraic-theory-expressible) why asymptotic/quantitative number theory resists categorification?