${\mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}}_R(\mathcal{V})$-enriched $\mathrm{\infty }$-categories are left $R$-module objects of $\mathcal{C}a{t}^{\mathcal{V}}$ and $\mathcal{C}a{t}^{\mathcal{V}}$-enriched $\mathrm{\infty }$-functors

We investigate ${\mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}}_R(\mathcal{V})$-enriched $\mathrm{\infty }$-categories, where $R$ is an ${\mathbb{E}}_2$-ring in a presentable ${\mathbb{E}}_2$-monoidal $\mathrm{\infty }$-category $\mathcal{V}$, using $\mathcal{V}$-enriched $\mathrm{\infty }$-category theory. We prove the equivalence of $\mathcal{C}a{t}_{\mathrm{\infty }}^{{\mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}}_R(\mathcal{V})}$ (the $\mathrm{\infty }$-category of ${\mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}}_R(\mathcal{V})$-enriched $\mathrm{\infty }$-categories) and ${\mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}}_R(\mathcal{C}a{t}_{\mathrm{\infty }}^{\mathcal{V}})$ (left $R$-modules in $\mathcal{C}a{t}_{\mathrm{\infty }}^{\mathcal{V}}$). For $R$ an ${\mathbb{E}}_2$-ring in a presentable ${\mathbb{E}}_3$-monoidal $\mathrm{\infty }$-category, they are also equivalent to $Fu{n}^{\mathcal{C}a{t}_{\mathrm{\infty }}^{\mathcal{V}}}(B^{2}R,\mathcal{C}a{t}_{\mathrm{\infty }}^{\mathcal{V}})$, 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 $\mathrm{\infty }$-category, $(\mathrm{\infty },n)$-categories enriched in ${\mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}}_R(\mathcal{V})$ are equivalent to $B^{n}R$-modules in $\mathcal{V}$-enriched $(\mathrm{\infty },n)$-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 $\mathrm{\infty }$-category of dg-categories over $k$, a key object in derived algebraic geometry.