$\mathrm{LMod}_{R}(\mathcal{V})$-enriched $\infty$-categories are left $R$-module objects of $\mathcal{C}at^{\mathcal{V}}$ and $\mathcal{C}at^{\mathcal{V}}$-enriched $\infty$-functors

Abstract

We investigate $\mathrm{LMod}_{R}(\mathcal{V})$-enriched $\infty$-categories, where $R$ is an $\mathbb{E}_2$-ring in a presentable $\mathbb{E}_2$-monoidal $\infty$-category $\mathcal{V}$, using $\mathcal{V}$-enriched $\infty$-category theory. We prove the equivalence of $\mathcal{C}at_{\infty}^{\mathrm{LMod}_{R}(\mathcal{V})}$ (the $\infty$-category of $\mathrm{LMod}_{R}(\mathcal{V})$-enriched $\infty$-categories) and $\mathrm{LMod}_{R}(\mathcal{C}at_{\infty}^{\mathcal{V}})$ (left $R$-modules in $\mathcal{C}at_{\infty}^{\mathcal{V}}$). For $R$ an $\mathbb{E}_2$-ring in a presentable $\mathbb{E}_3$-monoidal $\infty$-category, they are also equivalent to $Fun^{\mathcal{C}at_{\infty}^{\mathcal{V}}}(B^2R,\mathcal{C}at_{\infty}^{\mathcal{V}})$, where $B^2(-)$ is the "$2$-delooping". This result generalizes: if $R$ is an $\mathbb{E}_{n+1}$-ring in a presentable $\mathbb{E}_{n+1}$-monoidal $\infty$-category, $(\infty,n)$-categories enriched in $\mathrm{LMod}_{R}(\mathcal{V})$ are equivalent to $B^nR$-modules in $\mathcal{V}$-enriched $(\infty,n)$-categories, where $B^n(-)$ is the "$n$-delooping". A notable case is $\mathcal{V} = \mathcal{S}p$ and $R = \mathbb{H}\mathrm{k}$, the Eilenberg-MacLane spectrum of a commutative ring $k$. In this case, the results provide two new descriptions of $\mathcal{D}(k)$ the $\infty$-category of dg-categories over $k$, a key object in derived algebraic geometry.