Homological Perturbation Theory for Algebras over Operads

We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. Specifically, for an operad O, we define the notion of an ‘O-algebra contraction’ and we prove that the formulas of the Basic Perturbation Lemma preserve O-algebra contractions. Over a ground ring containing the rational numbers, we give explicit formulas for constructing an O-algebra contraction from any given contraction, generalizing the so called ‘Tensor Trick’. As an illustration of our results we use them to give short proofs of the transfer and minimality theorems for O∞-algebras, where O is any Koszul operad. This subsumes, but is not restricted to, the cases of A∞, C∞ and L∞-algebras.