Enriched $\infty$-categories as marked module categories
David Reutter, Markus Zetto
TL;DR
This work introduces a new, model-agnostic way to handle ∞-categories enriched in a presentably monoidal ∞-category $\eta V$ by encoding the enrichment data entirely in the enriched presheaf category together with a marking by representables. Central to the approach is the notion of marked modules, which makes the enrichment functorial in both the object-space and the enriching category, and enables a monadic reconstruction of enriched categories from presheaf data via a Barr–Beck–Lurie style theorem in presentable module categories. The paper proves a key equivalence: the category ${f vCat}_X(\mathcal V)$ of $\,\mathcal V$-enriched ∞-categories with object space $X$ embeds fully faithfully into the category of marked $\,\mathcal V$-modules, with the essential image consisting of those marked modules generated by atomic objects and subject to a suitable conservativity condition on Hom-spaces. It further develops univalence, flagged categories, and a tensor product of enriched categories, showing compatibility with colimits and establishing a robust monoidal framework that unifies several existing models (operadic, lax enrichment, and Day convolution). Overall, the marked-module perspective reduces enriched ∞-category theory to the higher algebra of presentable ∞-categories, clarifies functoriality, and provides a flexible, model-independent foundation with concrete applications to univalence and tensor products.
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.