  There is a construction which lies at the heart of descent theory. The
combinatorial aspects of this paper concern the description of the construction
in all dimensions. The description is achieved precisely for strict
n-categories and outlined for weak n-categories. The categorical aspects
concern the development of descent theory in low dimensions in order to provide
a template for a theory in all dimensions. The theory involves non-abelian
cohomology, stacks, torsors, homotopy, and higher-dimensional categories. Many
of the ideas are scattered through the literature or are folklore; a few are
new.
