  A Stein covering of a complex manifold may be used to realise its analytic
cohomology in accordance with the Cech theory. If, however, the Stein covering
is parameterised by a smooth manifold rather than just a discrete set, then we
construct a cohomology theory in which an exterior derivative replaces the
usual combinatorial Cech differential. Our construction is motivated by
integral geometry and the representation theory of Lie groups.
