In the theory of group actions on posets one studies the action of a group $G$ on a poset $\mathcal{P}$ (via order-preserving permutations) partly through its derived action on the order complex of the poset, $\Delta(\mathcal{P})$, which is a simplicial complex constructed from linear chains in the poset, and associated linear representations on the (co)homology of (geometric realizations of) $\Delta(\mathcal{P})$. (See R. Stanley's Some aspects of groups acting on finite posets, 1982, for example.)
In arithmetic we have a basic poset: the collection of all number fields, which is not a finite poset, or even locally finite, but does have an initial element, $\mathbb{Q}$. Call this poset $\mathcal{P}(\mathbb{Q})$. We also have analogous "local" posets $\mathcal{P}(\mathbb{Q}_p)$, one for each rational prime $p$. Each of these posets is acted on by the absolute Galois group of its initial element, $G_\mathbb{Q}$ or $G_{\mathbb{Q}_p}$.
Have these posets and associated simplicial/topological/linear Galois actions been studied in some guise, and if so, where, or is there an elementary error I am making in thinking along these lines? I suspect the latter because order complexes of these kinds of posets seem such natural topological gadgets to attach to number fields and their localizations and study Galois actions on, and yet one doesn’t come across them. Of course, it is also possible that an expert sees immediately that there isn’t likely to be any interesting information to be extracted from looking at these rather unwieldy infinitary objects that isn’t already available with simpler means.