In the 1950s, topologists introduced the notion of equivariant cohomology $H_G(E)$ for a topological space $E$ with an action by a compact group $G$. If the action is free, $H_G(E)$ should be $H(E/G)$, and be computed using de Rham cohomology. In 1950, even before the concept of equivariant cohomology had been formulated, Henri Cartan introduced a complex of equivariant differential forms for a compact Lie group acting on a differential manifold $E$, and proved a result amounting to stating that the cohomology of that complex computes $H_G(E)$. In 1999, Guillemin and Sternberg reformulated Cartan’s work in terms of a supersymmetric extension of the Lie algebra of $G$.

Our aim is to reconsider such considerations, by replacing vector spaces by a $k$-linear symmetric monoidal category, requiring that this category contain an odd unit to account for the supersymmetric dimension plus some further properties, and considering modules of a rigid Lie algebra object in that category. In that context, we obtain a version of Koszul’s homotopy isomorphism theorem, and recover as a consequence some known results as the acyclicity of the Koszul resolution. (Joint work with Siddhartha Sahi.)