Completion for braided enriched monoidal categories

Understand

Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are classified by braided oplax monoidal functors from $\mathcal{V}$ to the Drinfeld centers of ordinary monoidal categories.

• In this article, we prove that this classifying functor is strongly monoidal if and only if the original $\mathcal{V}$-monoidal category is tensored over $\mathcal{V}$.
• We then define a completion operation which produces a tensored $\mathcal{V}$-monoidal category $\overline{\mathcal{C}}$ from an arbitrary $\mathcal{V}$-monoidal category $\mathcal{C}$, and we determine many equivalent conditions which imply $\mathcal{C}$ and $\overline{\mathcal{C}}$ are $\mathcal{V}$-monoidally equivalent.
• Since being tensored is a property of the underlying $\mathcal{V}$-category of a $\mathcal{V}$-monoidal category, we begin by studying the equivalence between (tensored) $\mathcal{V}$-categories and oplax (strong) $\mathcal{V}$-module categories respectively.