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. We then define the completion operation for $\mathcal{ V}$-categories, and adapt these results to the $\mathcal{ V}$-monoidal setting.
Categories: math.CT math.QA