Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold–Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E∞-algebras. We prove that it in fact sends algebras over arbitrary differential graded E∞-operads to E∞-algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.