  A strong from of invariance under a group G is manifested in a family over
the classifying space BG. We advocate a differential-geometric avatar of BG
when G is a Lie group. Applied to G-equivariant connections on smooth principal
or vector bundles, the equivariance-->families principle converts the
G-equivariant extensions of curvature and Chern-Weil forms to the standard
nonequivariant versions. An application of this technique yields the moment map
of the determinant line of a G-equivariant Dirac operator, which in turn sheds
light on some anomaly formulas in quantum field theory.
