Re: question about monoidal categories

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

As an addendum to the interesting answers that have been
given so far to Paul's question, it is perhaps worth pointing
out that using an old result of Max Kelly's, the situation
Paul describes can be expressed purely in terms of a strong
monoidal functor.

Given a monoidal category M and functor F : M --> S as in
Paul's message, we define a strict monoidal category {F, F}
as follows. Its objects are triples (G, H, a) fitting into a
diagram of functors and natural transformations

        G
    M -----> M
    |        |
    |   a    |
F  |   =>   | F
    |        |
    v        v
    S -----> S
        H

Its morphisms (G, H, a) --> (G', H', a') are pairs of natural
transformations b : G => G', c : H => H' satisfying the
obvious compatibility condition with a and a'. The tensor
product is given by pasting squares next to each other
horizontally.

There are strict monoidal functors p_1 : {F, F} -> [M, M]
and p_2 : {F, F} -> [S, S] sending (G, H, a) to G and H
respectively; and as in my previous message we have strong
monoidal functors

R : M ---> [M, M]
     m |--> (-) * m

T : M ---> [S, S]
     m |--> id_S

corresponding to the right regular action of M on itself,
and to the trivial action of M on S.

Now to give the natural transformation beta of Paul's
message, satisfying his "monoidality" conditions, is
precisely to give a strong monoidal functor B: M --> {F, F}
rendering the diagram

           _  {F, F}
           .|   |
         .      |
     B .        | (p_1, p_2)
     .          |
   .            v
M --------> [M, M] x [S, S]
    (R, T)

commutative.

I believe this technique originates in the paper

"Coherence theorems for lax algebras and for distributive
laws", G.M. Kelly, LNM 420.

A good place to learn more about it is in Section 2 of

"On property-like structures", G.M. Kelly and S. Lack, TAC Vol. 3, No. 9



Richard


--On 07 February 2008 20:05 Paul B Levy wrote:

>
>
>
> Let F be a functor from a monoidal category M to a category S.
>
> We are given
>
>        beta(p,a) : F(p) --> F(p*a)
>
> natural in p,a in M.
>
> If I tell you that, in addition to naturality, beta is "monoidal", I'm
> sure you will immediately guess what I mean by this, viz.
>
> (a) for any p,a,b in M
>
>   beta(p,a) ; beta(p*a,b) = beta(p,a*b) ; F(alpha(p,a,b))
>
> (b) for any p in M
>
>   beta(p,1) = F(rho(p))
>
> Yet I cannot see any reason for giving the name "monoidality" to (a)-(b).
>
> It doesn't appear to be a monoidal natural transformation in the official
> sense.  There are no monoidal functors in sight.
>
> Can somebody please justify my usage?
>
> Paul
>
>
>