  Measuring comodules are defined and shown to provide a useful generalization
of the set of maps between modules with a broad range of applications. Three
applications are described. Connections on bundles are described in terms of
measuring comodules, enabling curvature to be defined under general algebraic
circumstances. Loop algebras are realized via a short exact sequence of
measuring comodules, with the central extension given by the curvature. Finally
dual comodules provide a method of dualizing representations, which, when
applied to representations of loop algebras yield positive energy
representations, and when applied to representations of totally disconnected
groups leads to the smooth dual.
