An equivalence between enriched ∞-categories and ∞-categories with weak action

Abstract

We show that an ∞-category M with a closed left action of a monoidal ∞-category V is completely determined by the V-valued graph of morphism objects resulting from closedness of the action equipped with the structure of a V-enrichment in the sense of Gepner-Haugseng. We prove a similar result when M is a V-enriched ∞-category in the sense of Lurie, an operadic generalization of the notion of ∞-category with closed action. Precisely, we prove that sending a V-enriched ∞-category in the sense of Lurie to the V-valued graph of morphism objects refines to an equivalence χ between the ∞-category of V-enriched ∞-categories in the sense of Lurie and of Gepner-Haugseng.Moreover if V is a presentably k+1-monoidal ∞-category for 1≤k≤∞, we prove that χ restricts to a lax k-monoidal functor between the ∞-category of left V-modules in PrL, the symmetric monoidal ∞-category of presentable ∞-categories, endowed with the relative tensor product, and the tensor product of V-enriched ∞-categories of Gepner-Haugseng.As an application of our theory we construct a lax symmetric monoidal embedding of the ∞-category of small stable ∞-categories into the ∞-category of small spectral ∞-categories. As a second application we produce an enriched Yoneda-embedding in the framework of Lurie's notion of enriched ∞-categories.