Transferring algebra structures up to homology equivalence

Given a strong deformation retract $M$ of an algebra $A$, there are several apparently distinct ways ([9],[19],[13],[24],[15],[18],[17]) of constructing a coderivation on the tensor coalgebra of $M$ in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [17] bar construction of $A$. We show that these methods are equivalent and are determined combinatorially by an inductive formula first given in a very special setting in [16]. Applications to de Rham theory and Massey products are given.