Relative operads model $\infty$-operads

Abstract

Given a (colored) operad and a set of unary operations, we can form an associated $\infty$-operad via localization. We show that localization determines an equivalence of homotopy theories of relative operads and $\infty$-operads. As an application, we give an affirmative answer to an open question by Harpaz, proving that Lurie's operadic nerve functor determines an equivalence of homotopy theories of simplicial operads and Lurie's $\infty$-operads.