  The notion of a generalized product, refining that of a (symmetric and
smooth) simplicial space is introduced and shown to imply the existence of an
algebra of pseudodifferential operators. This encompasses many constructions of
such algebras on manifolds with corners. The main examples discussed in detail
here are related to the semiclassical (and adiabatic) calculus as used in the
approach to a twisted form of the Atiyah-Singer index theorem in work with Is
Singer and Mathai Varghese.
