Monoidal Categories Enriched in Braided Monoidal Categories

We introduce the notion of a monoidal category enriched in a braided monoidal category V. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld centre of some monoidal category 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. Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors V → Z(T). We would like to understand this further; in a future article, we show that the functor is strong if and only if the enriched category is 'complete' in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like. One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor Rep(G) → Z(T) for some finite group G and a monoidal category T, and produces a new monoidal category T_||G. In our setting, given any braided oplax monoidal functor V → Z(T), for any braided V, we produce T_||V: this is not usually an 'honest' monoidal category, but is instead V-enriched. If V has a braided lax monoidal functor to Vec, we can use this to reduce the enrichment to Vec, and this recovers de-equivariantization as a special case.