An $A_{\mathrm{\infty }}$-version of the Eilenberg-Moore theorem

We construct an $A_{\mathrm{\infty }}$-structure on the two-sided bar construction involving homotopy Gerstenhaber algebras (hgas). It extends the non-associative product defined by Carlson and the author and generalizes the dga structure on the one-sided bar construction due to Kadeishvili-Saneblidze. As a consequence, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is promoted to a quasi-isomorphism of $A_{\mathrm{\infty }}$-algebras. We also show that the resulting product on the differential torsion product involving cochain algebras agrees with the one defined by Eilenberg-Moore and Smith, for all triples of spaces. This is a consequence of the following result, which is of independent interest: The strongly homotopy commutative (shc) structure on cochains inductively constructed by Gugenheim-Munkholm agrees with the one previously defined by the author for all hgas.