Enriched $\infty$-operads as marked algebras

Abstract

We show that an enriched $\infty$-operad is completely determined by its category of right modules together with a `marking' of the representable modules. More precisely, for any presentably monoidal $\infty$-category $\mathcal{V}$ we construct an equivalence between the category of colored $\mathcal{V}$-enriched $\infty$-operads and a certain full subcategory of the category of presentably symmetric monoidal $\mathcal{V}$-module $\infty$-categories equipped with a functor from an $\infty$-groupoid. This effectively allows us to reduce many aspects of enriched $\infty$-operad theory to the theory of presentably symmetric monoidal $\infty$-categories. As an application, we describe a notion of univalence (or Rezk-completeness) for enriched $\infty$-operads, and directly construct an equivalence between univalent $\mathcal{S}$-enriched $\infty$-operads in our sense and Lurie's model of $\infty$-operads. We study envelopes and categories of algebras for enriched $\infty$-operads and show that, in the $\mathcal{S}$-enriched case, the resulting notions agree in both models.