$\mathrm{Mod}_{\mathbb{H}\mathrm{k}}$-enriched $\infty$-categories are left $\mathbb{H}\mathrm{k}$-module objects of $\mathcal{C}at_{\infty}^{\mathcal{S}p}$ and $\mathcal{C}at_{\infty}^{\mathcal{S}p}$-enriched $\infty$-functors

Abstract

We establish the feasibility of investigating the theory of $\mathrm{Mod}_{\mathbb{H}\mathrm{k}}$-enriched $\infty$-categories, where $\mathbb{H}\mathrm{k}$ is the Eilenberg-Maclane Spectrum associated with a commutative and unitary ring $k$, through the framework of $\mathcal{S}p$-enriched $\infty$-category theory. In particular, we prove that the $\infty$-category of $\mathrm{Mod}_{\mathbb{H}\mathrm{k}}$-enriched $\infty$-categories $\mathcal{C}at_{\infty}^{\mathrm{Mod}_{\mathbb{H}\mathrm{k}}}$, $\infty$-category of left $\mathbb{H}\mathrm{k}$-module objects of the $\infty$-category of $\mathcal{S}p$-enriched $\infty$-categories $\mathcal{C}at_{\infty}^{\mathcal{S}p}$ $\mathrm{LMod}_{\mathbb{H}\mathrm{k}}(\mathcal{C}at_{\infty}^{\mathcal{S}p})$ and the $\infty$-category of $\mathcal{C}at_{\infty}^{\mathcal{S}p}$-enriched $\infty$-functors $Fun^{\mathcal{C}at_{\infty}^{\mathcal{S}p}}(\underline{\underline{\mathbb{H}\mathrm{k}}},\mathcal{C}at_{\infty}^{\mathcal{S}p})$ are equivalent.