Moduli problems for operadic algebras

A theorem of Pridham and Lurie provides an equivalence between formal moduli problems Lie algebras in characteristic zero. We prove a generalization this correspondence, relating parametrized by over Koszul operad to its dual operad. In particular, when the algebra associated deformation problem is induced from pre-Lie structure it corresponds permutative problem. As another example we obtain correspondence operadic augmented operads.