Equivariant cohomology and tensor categories

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.