Resolution of Coloured Operads and Rectification of Homotopy Algebras

Abstract

We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model struc- ture, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their cat- egories of algebras. We construct an explicit Boardman-Vogt style cofibrant resolution for coloured operads, thereby giving a uniform approach to algebraic structures up to homotopy over coloured operads. The Comparison Theorem implies that such structures can be rectified.