Enriched $\infty$-categories as marked module categories

TL;DR

This paper establishes a new equivalence between enriched ∞-categories and certain presentable module categories, enabling a reduction of enriched ∞-category theory to the theory of presentable ∞-categories, with applications to tensor products and univalence.

Contribution

It introduces an equivalence linking enriched ∞-categories to presentable module categories with markings, simplifying the study of enriched ∞-categories and enabling new constructions.

Findings

Constructed a tensor product of enriched ∞-categories with desirable properties

Reformulated univalence for enriched ∞-categories in a model-independent way

Proved a monadicity theorem for presentable module categories

Abstract

We prove that an enriched $\infty$-category is completely determined by its enriched presheaf category together with a `marking' by the representable presheaves. More precisely, for any presentably monoidal $\infty$-category $\mathcal{V}$ we construct an equivalence between the category of $\mathcal{V}$-enriched $\infty$-categories and a certain full sub-category of the category of presentable $\mathcal{V}$-module categories equipped with a functor from an $\infty$-groupoid. This effectively allows us to reduce many aspects of enriched $\infty$-category theory to the theory of presentable $\infty$-categories. As applications, we use Lurie's tensor product of presentable $\infty$-categories to construct a tensor product of enriched $\infty$-categories with many desirable properties -- including compatibility with colimits and appropriate monoidality of presheaf functors -- and compare it to existing tensor products in the literature. We also re-examine and provide a model-independent reformulation of the notion of univalence (or Rezk-completeness) for enriched $\infty$-categories. Our comparison result relies on a monadicity theorem for presentable module categories which may be of independent interest.