monoidal enriched natural transformations

From: Agustí Roig (soquiso_at_hotmail.com)
Date: 10/28/04 

Date: Thu, 28 Oct 2004 20:30:08 +0000 (UTC)

Hi.

I have a problem which, it seems to me, requires the notion of
enriched natural transformation between enriched monoidal functors,
but I haven't been able to find a good reference for it.

I've taken a look at the books of Borceux (Handbook of categorical
algebra), Kelly (Basic notions of enriched category) and the appendix
in Levine's "Mixed motives", but they all end just where I need them,
or even long before that point.

More precisely, the situation I've encountered seems to be the
following. I have:

- Two monoidal (symmetric) categories:

        V and W .

- A couple of monoidal (symmetric) functors between them:

        S, T : V ---> W

- A monoidal natural transformation between S and T :

        \omega : S ===> T

And here is where my problems begin.

There is a well-known notion of what is a V-functor between
V-categories and what a V-natural transformation is.

My first need is to understand what an S-functor between a
V-category C and a W-category D should be:

        F : C ---> D

I didn't find this thing in the literature, but I expect it ought to
be something like a V-functor, but with a family of morphisms in W

        \lambda_{XY} : S[X,Y] ---> [FX,FY]

for every pair of objects X, Y in C . (Here the square brackets
[,] stand for the objects in V and W of "morphisms" of C and D
, and I'm leaving aside units and commutative isomorphisms for the
moment.)

This seems reasonable to me, since (a) is the situation I have in the
"real" world and (b) if I put S = id_V , I find the definition of a
V-functor.

Next, I would need the notion of an "\omega - natural transformation"
and I think this should be something like a V-natural transformation
between an S-functor F : C ---> D and a T-functor G: C ---> D,
but placing at the beginning of the commutative diagram which defines
a V-natural transformation an arrow like

        \omega_{[X,Y]} : S[X,Y] ---> T[X,Y] .

Assuming that this is ok, I should also need to understand what might
be the definition of a "monoidal \omega - natural transformation".

That is to say, C is a monoidal V-category, D is a monoidal
W-category, F is a monoidal S-functor and G a monoidal T-functor:
what is a monoidal natural transformation between F and G , over
the monoidal natural transformation \omega : S ===> T ?

I've drawn a couple of commutative diagrams that should appear in the
definition of such a construct, but I feel I could be forgetting a
dozen more. Any references for it?

Unfortunately for me, Kelly's book ends before this point: it
explicitely says: "is our decision not to discuss the 'change of
base-category' given by a symmetric monoidal functor V ---> W". Has
someone else done the job after Kelly's book?

Agustí Roig