We describe an organizing framework to study higher dimensional infinitary combinatorics based on \v{C}ech cohomology, originating from works by Barry Mitchell, Barbara Osofsky and others. A central combinatorial notion is n-dimensional coherence sequences, generalizing the 1-dimensional ones studied extensively by Todorcevic using the method of minimals walks. We will discuss ZFC results suggesting aleph_n is not "compact for n+1-dimensional combinatorics" and consistency results that any regular cardinal greater or equal to aleph_{omega+1} can be "compact for n-dimensional combinatorics for all n". The talk will be purely combinatorial. Joint work with Jeffrey Bergfalk and Chris Lambie-Hanson.