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We characterize $A_\infty$-structures that are transfers over a chainhomotopy 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 ourresults to ${\mathcal P}_\infty$-structures over a field of characteristiczero, for any quadratic Koszul operad ${\mathcal P}$.

Which homotopy algebras come from transfer?