Boardman Vogt W construction for modules over an operad from Top Questions - Math Overflow by Jeremy Miller The W construction of Boardman and Vogt gives a cofibrant replacement for operads. In http://arxiv.org/abs/math/9907073, Salvatore describes a cofibrant replacement for algebras over an operad. Is there a similar construction which produces cofibrant resolutions of right or left modules over an operad? Does anyone have know of a reference for this? http://mathoverflow.net/questions/97701/boardman-vogt-w-construction-for-modules-over-an-operad
arXiv:1205.6058 Homotopy unital A_infinity-algebras from arXiv Front: math.AT by Volodymyr Lyubashenko It is well known that the differential graded operad of A_infinity-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_infinity-algebras. We prove that the operad bimodule of A_infinity-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_infinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras with units.
nLab page on Operad II