Homotopy unital A∞-algebras

It is well known that the differential graded operad of A∞-algebras is a cofibrant replacement (a dg-resolution) of the operad of associative differential graded algebras without units. In this article we find a cofibrant replacement of the operad of associative differential graded algebras with units. Algebras over it are called homotopy unital A∞-algebras. We prove that the operad bimodule of A∞-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras without units. Similarly we show that the operad bimodule of homotopy unital A∞-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras with units.