An equivalence between enriched $\infty$-categories and $\infty$-categories with weak action

Hadrian Heine

Published 2020-09-05Version 1

We show that an $\infty$-category $\mathcal{M}$ left tensored over a monoidal $\infty$-category $\mathcal{V}$ is completely determined by its graph $$\mathcal{M}^\simeq \times \mathcal{M}^\simeq \to \mathcal{P}(\mathcal{V}), \ (\mathrm{X},\mathrm{Y}) \mapsto \mathcal{M}((-)\otimes\mathrm{X},\mathrm{Y}),$$ parametrized by the maximal subspace $\mathcal{M}^\simeq$ in $\mathcal{M}$, equipped with the structure of an enrichment in the sense of Gepner-Haugseng in the Day-convolution monoidal structure on the $\infty$-category $\mathcal{P}(\mathcal{V})$ of presheaves on $\mathcal{V}.$ Precisely, we prove that sending an $\infty$-category left tensored over $\mathcal{V}$ to its graph defines an equivalence between $\infty$-categories left tensored over $\mathcal{V}$ and a subcategory of all $\infty$-categories enriched in presheaves on $\mathcal{V}.$ More generally we consider a generalization of $\infty$-categories left tensored over $\mathcal{V}$, which Lurie calls $\infty$-categories pseudo-enriched in $\mathcal{V}$, and extend the former equivalence to an equivalence $\chi$ between $\infty$-categories pseudo-enriched in $\mathcal{V}$ and all $\infty$-categories enriched in presheaves on $\mathcal{V}.$ The equivalence $\chi$ identifies $\mathcal{V}$-enriched $\infty$-categories in the sense of Lurie with $\mathcal{V}$-enriched $\infty$-categories in the sense of Gepner-Haugseng. Moreover if $\mathcal{V}$ is symmetric monoidal, we prove that sending an $\infty$-category left tensored over $\mathcal{V}$ to its graph is lax symmetric monoidal with respect to the relative tensorproduct on $\infty$-categories left tensored over $\mathcal{V}$ and the canonical tensorproduct on $\mathcal{P}(\mathcal{V})$-enriched $\infty$-categories.