Homotopy unital A∞-algebras

Abstract 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.