The root functor

In this paper, we show that any $\mathrm{\infty }$-operad is equivalent to the localization of a discrete $\mathrm{\Sigma }$-free operad; this result extends Joyal’s delocalization theorem for categories to the operadic setting. Along the way, we pursue a systematic study of $\mathrm{\infty }$-operadic localization in the dendroidal context and its compatibility with un/straightening equivalences, deducing another description of algebras over $\mathrm{\infty }$-operads.