DOCUMENTA MATHEMATICA, Vol. 20 (2015), 879-926

Let $\cO$ be a topological (colored) operad. The Lurie $\infty$-category of $\cO$-algebras with values in ($\infty$-category of) complexes is compared to the $\infty$-category underlying the model category of (classical) dg $\cO$-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.