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Keywords:
Eilenberg–Moore theorem; bar construction; A_$\infty$-algebra; homotopy Gerstenhaber algebra; shc algebra
Eilenberg–Moore theorem; bar construction; A_$\infty$-algebra; homotopy Gerstenhaber algebra; shc algebra
Summary:
We construct an A_$\infty$-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_$\infty$-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.
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