  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.
  Strikingly, our classification is slightly more general than what one might
have anticipated in terms of strong monoidal functors $\mathcal V \to
Z(\mathcal T)$. We would like to understand this further; in a future paper 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 $\mathsf{Rep}(G)
\to Z(\mathcal T)$ for some finite group $G$ and a monoidal category $\mathcal
T$, and produces a new monoidal category $\mathcal T // G$. In our setting,
given any braided oplax monoidal functor $\mathcal V \to Z(\mathcal T)$, for
any braided $\mathcal V$, we produce $\mathcal T // \mathcal V$: this is not
usually an `honest' monoidal category, but is instead $\mathcal V$-enriched. If
$\mathcal V$ has a braided lax monoidal functor to $\mathsf{Vec}$, we can use
this to reduce the enrichment to $\mathsf{Vec}$, and this recovers
de-equivariantization as a special case.
