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The purpose of this paper is twofold. First, we review applications of thebar duality of operads to the construction of explicit cofibrant replacementsin categories of algebras over an operad. In view toward applications, we checkthat the constructions of the bar duality work properly for algebras overoperads in unbounded differential graded modules over a ring.  In a second part, we use the operadic cobar construction to define explicitcyclinder objects in the category of operads. Then we apply this constructionto prove that certain homotopy morphisms of algebras over operads areequivalent to left homotopies in the model category of operads.

Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads