Re: Enrichment over a monoidal bicategory

[Steve Lack <s.lack@uws.edu.au>]

Dear Alex,

As you say, it is not hard to define bicategories enriched in a monoidal
bicategory; in fact the only hard thing is saying what a monoidal bicategory
is. As you also point out, these are quite different to categories enriched
in a bicategory, in the sense of Walters. The latter are still "strict"
structures; indeed they are categorical rather than 2-categorical, so there
is no room for any non-strictness.

Benabou [Introduction to bicategories, SLN 47] defined a polyad in a
bicategory B to be a set X equipped with a morphism of bicategories
X_ch-->B, where X_ch is the bicategory with object-set X and with all
hom-categories terminal. This is exactly what Walters later called a
B-enriched category, and used in his study of sheaves. (Benabou gave
categories enriched in a monoidal category as an example of polyads, but did
not explicitly suggest that polyads were a sort of enriched category.)

Gordon, Power, and Street [Coherence for tricategories, AMS Memoirs]
considered the next dimension up. For a tricategory T, they called a
morphism of tricategories X_ch-->T a T-category, although did not go on to
use this notion in any way. The case where T has one object is exactly
the situation you discuss.

There is a certain amount of flabbiness in this notion of T-categories,
coming, for example, from the use of not necessarily normal homomorphisms.
A tighter, more explicit definition of bicategories enriched in monoidal
bicategories was given by Sean Carmody in his 1995 Cambridge thesis. They
also appeared in my thesis the following year.

More recently, there has been quite a lot of work done on the one-object
case: pseudomonoids in Gray-monoids, or equivalently pseudomonads in
Gray-categories.

Hope this helps.

Steve Lack.


On 20/05/09 1:10 PM, "Alex Hoffnung" <alex@math.ucr.edu> 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
>
>