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.
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