The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the theory of formal groups, local fields and moduli spaces. In another one there is a whole subject of derived algebraic geometry, as well as more esoteric objects like chiral homology and their applications. Perhaps this is wishful thinking, but I would really like to view these two fields as different facets of some deeper theory.
In particular, since the days of yore does AT employ a theory of localizations and completions at ideals of $\mathbb Z$. This already makes it feel as if we're really doing algebraic geometry over $\mathrm{Spec}\ \mathbb Z$. This feeling is supported by Nishida's theorem, which tells us that the sphere spectrum can be really thought about as some very nice nilpotent thickening of $\mathbb Z$. With a bit of white magic we can even apply local class field theory to the study of homotopy groups. However, there is a glaring issue: in number theory we really should be working not over $\mathbb Z$, but over some compactification of it, and should include information at infinite primes. Surprisingly, I have never seen infinite primes mentioned in homotopy theory.
An obvious guess that we should study localizations w.r.t. $H\mathbb R$ instead of $H\mathbb Q$ or $H\mathbb{F}_p$ fails for a simple reason that rational cohomology 'equal' real and complex ones. Trying to google for something like "infinite primes in algebraic topology" or "infinite prime Bousfield localization" returned zero relevant results. My knowledge about infinite primes in number theory is very limited, but as far as I understand constructions mostly revolve around completions with respect to archimedean norms on extensions of $\mathbb Q$ and studying real vector bundles with metric (well, it's mostly the same). I see no way to push either of these approaches to homotopy (metric cohomology? wut?).
Thus the question as stated: is there any theory exploiting some constructions (especially some form of localization) with respect to infinite primes in number fields to gain homotopy-theoretic results? Some variant of topological Arakelov geometry would be close to the ideal result, but I don't expect that it exists, so would welcome any leading threads. More broadly, what could take the role of those infinite primes and supply the missing (in what sense?) homotopy-theoretic information?
Of course, one could note that perhaps a more basic question is how the Galois extensions of $\mathbb Q$ can be generalized to spectra (because otherwise there are not many potential infinite primes around), but I don't feel like it should really be an obstruction. In any case it should be a subject of another question.
Best Answer
The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).
I had an idea at some point (with the explicit thought that it could be something like Bousfield-Kan completion at the infinite place) of doing the same construction with the functor $S\mapsto B_1(\ell^1(S))$ in place of $\mathbb F_p[-]$. This makes sense because $B_1(\ell^1(-))$ is also a monad.
Here, $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.
I never pursued that idea. Tilman Bauer and I discussed it at some point, and we had the vague though that this might be related to the concept of $\ell^1$-homology. [Note that $\ell^1$-homology only sees $\pi_1$ and is completely insensitive to higher homotopy groups (and moreover, it vanishes identically when $\pi_1$ is ameanable), so it belongs more to the area of geometric group theory and less to algebraic topology.]