Layer cake and homotopy representations I: formal geometry approach

Representations up to homotopy of Lie algebras have attracted recently much attention. On the other hand J. Baez has introduced a way to build a homotopy Lie algebra out of a Lie algebra and an n-cocycle. We show in this work a common framework enabling to generalize both notions (replacing Lie algebras by homotopy Lie algebras) and extend them for other types of algebras (commutative and associative). The main tool is the language of homological vector fields on products of formal manifolds. This is a joint work with John Baez.