Enrichment over a monoidal bicategory

[Alex Hoffnung <alex@math.ucr.edu>]

Hi

I have found that there is a fairly straightforward way to generalize
the notion of enrichment over a monoidal category to enrichment over a
monoidal bicategory.  Namely, a "bicategory enriched over a monoidal
bicategory V" consists of the following:

1) a collection of "objects" A, B, C,...

2) for any pair of objects A,B, an object in V called hom(A,B)

3) for any triple of objects A,B,C a morphism in V called composition:
hom(A,B) tensor hom(B,C) -> hom(A,C)
where "tensor" is the tensor product in V.

4) for any object A a morphism in V called identity: I_A -> hom(A,A)

5) for any quadruple of objects A,B,C,D a 2-isomorphism in V called
the associator, which does the obvious thing.

plus left and right unitors, and so on with all the axioms closely
following those of the definition of a bicategory.

I am looking to be pointed in the right direction in the literature.
Can anyone help?  I am aware of the fc-multicategories by Leinster and
earlier work by Walters, but those do not seem to use the monoidal
structure to enrich as I want.

Best,
Alex Hoffnung