Dear Alex, A fair amount of the theory of enriched bicategories is worked out in Steve Lack's PhD thesis "The algebra of distributive and extensive categories". I don't think there have been any further attempts to develop the theory to any serious degree. Best wishes, Richard --On 19 May 2009 22:10 Alex Hoffnung wrote:
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