I'm sure it depends on what you mean by "homotopy theorist-friendly", but I was kind of shocked to realize that even an "algebraic-geometer-friendly" approach to number theory doesn't seem to exist, at least in the sense I'd understand it.
And the intimate relationship between these fields has been much more widely known for a much longer time than the relationship between homotopy theory and number theory.
I have a question related to the above discussion about homotopy co/finality: is there a good category theoretic (i.e. doesn't mention weakly contractible nerve) way to get at homotopy cofinal functors (where I mean the ones along whose pullbacks colimits do not change)?
A functor $u: I \to J$ is (co)final for any functor $f: J \to \mathcal{S}$ the canonical map $\varinjlim fu \to u\varinjlim f$ is an isomorphism, where $\mathcal{S}$ refers to the infinity category of spaces.
