Rectification of algebras and modules

Abstract

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