Classifying spaces of algebras over a prop

Abstract

We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop creates difficulties and we introduce new methods to overcome them. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes.