We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\hat{ \mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\hat{ \mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\hat{ \mathcal O}$ in terms of derivations and module maps. An application to Poincar\'e duality on the chain level of a suitable topological space is given.
Categories: math.AT