  The point of departure for the present work is Barry Mitchell's 1972 theorem
that the cohomological dimension of $\aleph_n$ is $n+1$. We record a new proof
and mild strengthening of this theorem; our more fundamental aim, though, is
some clarification of the higher-dimensional infinitary combinatorics lying at
its core. In the course of this work, we describe simplicial characterizations
of the ordinals $\omega_n$, higher-dimensional generalizations of coherent
Aronszajn trees, bases for critical inverse systems over large index sets,
nontrivial $n$-coherent families of functions, and higher-dimensional
generalizations of portions of Todorcevic's walks technique. These
constructions and arguments are undertaken entirely within a $\mathsf{ZFC}$
framework; at their heart is a simple, finitely iterable technique of
compounding $C$-sequences.
