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 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 left 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 between the ∞-category of V-enriched ∞-categories in the sense of Lurie and of Gepner-Haugseng. Moreover if V is a presentably Ek+1-monoidal ∞-category for 1 ≤ k ≤ ∞, we prove that restricts to a lax Ek-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 a Yoneda-embedding for Lurie's notion of enriched ∞-categories.