Relative dendroidal Rezk nerve and applications

Abstract

We extend the dendroidal Rezk nerve to the setting of relative $\infty$-operads. Our main theorem relates it to localization of $\infty$-operads, generalizing a theorem of Mazel-Gee. By exploiting the relation, we obtain a surprisingly effective tool to prove localization results in operadic contexts. As applications, we obtain a number of new results on operadic localizations, including a generalization of Willwacher's recent result on cyclic operads and operadic modules, and a description of locally constant factorization algebras on spheres in terms of discrete geometry.