We describe an organizing framework for the study of infinitary combinatorics. This framework is Cˇech cohomology. It describes ZFC combinatorial principles distinguishing among higher ωn. More precisely, it correlates each ωn with an (n + 1)-dimensional generalization of Todorcevic’s walks technique, and begins to explain that technique’s "unreasonable effectiveness" on ω1. We show in contrast that on higher cardinals κ, the existence of these principles is frequently independent of the ZFC axioms. Finally, we detail implications of these phenomena for the computation of strong homology groups and higher derived limits, deriving independence results in algebraic topology and homological algebra, respectively, in the process.