Re: question about monoidal categories

[Sam Staton <ss368@cam.ac.uk>]

Hello, here is an answer to Paul's question: The data (F,beta)
defines a (lax?) functor between "monoidal categories with right
units but not left units" [henceforth MR-categories].

Every pointed category can be considered as an MR-category, in fact a
strict one. The tensor product is left projection. The right unit is
the point of the category (although every object of the category
behaves like a right unit for this category.)

In particular, the category S in Paul's email, with point F(1), can
be seen as an MR-category. Of course, every monoidal category,
including M, is an MR-category too.

The data for a lax MR-functor M->S consists of a functor F:M->S, a
morphism F(i)->F(i), which we can take as identity, and a natural
transformation
    F(p)*F(a) --> F(p*a)
which in this case amounts to a map
    beta:F(p) --> F(p*a)

A monoidal functor must satisfy three coherence conditions,
for associativity, left identity and right identity.
For an MR-functor, there are only axioms for associativity and right
identity, and these are exactly the axioms that Paul gave.

Hope that makes sense! All the best, Sam.


On 7 Feb 2008, at 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
>
>