A cohomological framework for homotopy moment maps

Given a Lie group acting on a manifold MM preserving a closed n+1n+1-form ωω, the notion of homotopy moment map for this action was introduced in Fregier (0000), in terms of L∞L∞-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of the results of Fregier (0000).