The Boardman–Vogt resolution of operads in monoidal model categories

Abstract

We extend the $W$-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed $\Sigma$-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar–bar chain resolution are shown to be particular instances of this generalised $W$-construction.