Enriched $\mathrm{\infty }$-operads as marked algebras

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