Consider a monoidal 2-category (or bicategory) B. For example, B could by the 2-category (finite sets, finite correspondences, isomorphisms of correspondences) with monoidal structure given by product of sets (etc). By a category C enriched over B I mean:
- A collection of objects ob(C)
- For each pair of objects x,y in ob(C), an object $Hom(x,y)$ in B.
- For each triple of objects x,y,z in ob(C), a morphism $Hom(y,z) \times Hom(x,y) \to Hom(x,z)$ in B.
- For each quadruple of objects x,y,z,w in ob(C), an isomorphism between the two ways of composing $Hom(z,w) \times Hom(y,z) \times Hom(x,y) \to Hom(x,w)$.
- Some units
satisfying some hypotheses.
Equivalently, we can view the monoidal 2-category B as a 3-category BB with a single object. Then a category enriched over B is the same as:
- A category C so that for each pair objects x,y, $Hom(x,y)$ has a single element. Promote C to a 3-category with only identity 2- and 3-morphisms.
- A 3-functor $C \to BB$.
There is also an interpretation in terms of multicategories.
Has this notion appeared somewhere before, and if so where? Note that this seems not to be the same as what nLab calls a category enriched in a bicategory (which is what Benabou calls a polyad).