Rectification of algebras and modules

Abstract

Let O be a topological (colored) operad. The Lurie ∞- category of O-algebras with values in (∞-category of) complexes is compared to the ∞-category underlying the model category of (classical) dg O-algebras. This can be interpreted as a "rectification" result for Lurie operad algebras. A similar result is obtained for modules over operad algebras, as well as for algebras over topological PROPs.