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