∞-OPERADS AS SYMMETRIC MONOIDAL ∞-CATEGORIES

Abstract

We use Lurie’s symmetric monoidal envelope functor to give two new descriptions of $\mathrm{\infty }$-operads: as certain symmetric monoidal $\mathrm{\infty }$-categories whose underlying symmetric monoidal $\mathrm{\infty }$-groupoids are free, and as certain symmetric monoidal $\mathrm{\infty }$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\mathrm{\infty }$-operads, as a localization of a presheaf $\mathrm{\infty }$-category, and we use this to give a simple proof of the equivalence between Lurie’s and Barwick’s models for $\mathrm{\infty }$-operads.