Which homotopy algebras come from transfer?

Abstract: We characterize $A_\infty$-structures that are transfers over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak $A_\infty$-morphisms over an arbitrary commutative ring. We then generalize our results to ${\mathcal P}_\infty$-structures over a field of characteristic zero, for any quadratic Koszul operad ${\mathcal P}$.