The root functor

Abstract

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

Version 2: major expository changes; section 5 moved to future work. Comments welcome!