The Cohomology of the Ordinals I: Basic Theory and Consistency Results
In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is Cech cohomology. We show in particular that the Cech cohomology groups of the ordinals articulate higher-dimensional generalizations of Todorcevic's walks and coherent sequences techniques, and begin to account for those techniques' `unreasonable effectiveness' on $\omega_1$. This discussion occupies the first half of our paper and is written with a general mathematical audience in mind. 
We turn in the paper's second half to more properly set-theoretic considerations. We describe a number of consistency results on the cohomology groups of the ordinals which certify their status as a graded family of incompactness principles. We show in particular that nontrivial cohomology groups on the ordinals are in some tension with large cardinals, and are maximally extant in Godel's model $\mathrm{L}$. We describe forcings to add, then trivialize, nontrivial $n$-cocycles, and conclude with some comparison of these principles with those benchmark incompactness phenomena, the existence of square sequences and failures of stationary reflection.