Re: Enrichment over a monoidal bicategory

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

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
>
>
>