Recent work of Ko Aoki has shown that the nonvanishing of sheaf cohomology groups of certain so-called locally profinite sets has important consequences in homological algebra and descent theory. Furthermore, for these spaces the statement "$H^n(X,\underline{\mathbb{Z}_2}) \neq 0$" admits a set-theoretic reformulation which is tractable by elementary arguments. In this talk we present techniques for computing these groups, based on an analysis of the degrees to which the corresponding spaces satisfy a principle reminiscent of Fubini's theorem. Time permitting, we will discuss applications to Clausen and Scholze's "light condensed mathematics".