  We propose the notion of a supercategory as an alternative approach to
supermathematics. We show that this setting is rich to carry out many of the
basic constructions of supermathematics. We also prove generalizations of a
number of results in equivariant cohomology, including the Chern-Weil theorem
for an arbitrary rigid Lie algebra object. For a quadratic Lie algebra object
we obtain a proof of the Duflo isomorphism along the lines of
Alekseev-Meinrenken, thereby generalizing their result to Lie superalgebras.
