An equivalence between enriched $\infty$-categories and $\infty$-categories with weak action

Abstract

We show that an $\infty$-category $\mathcal{M}$ with a closed left action of a monoidal $\infty$-category $\mathcal{V}$ is completely determined by the $\mathcal{V}$-valued graph of morphism objects equipped with the structure of a $\mathcal{V}$-enrichment in the sense of Gepner-Haugseng. We prove a similar result when $\mathcal{M}$ is a $\mathcal{V}$-enriched $\infty$-category in the sense of Lurie, an operadic generalization of the notion of $\infty$-category with closed left action. Precisely, we prove that sending a $\mathcal{V}$-enriched $\infty$-category in the sense of Lurie to the $\mathcal{V}$-valued graph of morphism objects refines to an equivalence $\chi$ between the $\infty$-category of $\mathcal{V}$-enriched $\infty$-categories in the sense of Lurie and of Gepner-Haugseng. Moreover if $\mathcal{V}$ is a presentably $\mathbb{E}_{\mathrm{k+1}}$-monoidal $\infty$-category for $1 \leq k \leq \infty$, we prove that $\chi$ restricts to a lax $\mathbb{E}_{\mathrm{k}}$-monoidal functor between the $\infty$-category of left $\mathcal{V}$-modules in $\mathrm{Pr}^L$, the symmetric monoidal $\infty$-category of presentable $\infty$-categories, endowed with the relative tensor product, and the tensor product of $\mathcal{V}$-enriched $\infty$-categories of Gepner-Haugseng. As an application of our theory we construct a lax symmetric monoidal embedding of the $\infty$-category of small stable $\infty$-categories into the $\infty$-category of small spectral $\infty$-categories. As a second application we produce a Yoneda-embedding for Lurie's notion of enriched $\infty$-categories.