  By expressing the geometric realization of simplicial sets and cyclic sets as
filtered colimits, Drinfeld (arXiv:math/0304064v3) proved in a substantially
simplified way the fundamental facts that geometric realization preserves
finite limits, and that the group of orientation-preserving homeomorphisms of
the interval [0,1] (resp. the circle R/Z) acts on the realization of a
simplicial (resp. cyclic) set. In this paper, we first review Drinfeld's method
and then introduce an analogous expression for the geometric realization of
dihedral sets. We also see how these expressions lead to a clarified
description of subdivisions of simplicial, cyclic, and dihedral sets.
