Monoidal categories enriched in braided monoidal categories

Understand

We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$.

• We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$.
• Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features.
• Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation.