Hi Paul, I think that I have written previously to the list about the possibility of a monoidal functor acting on a mere functor, and what you have is an instance of this notion. Here the monoidal functor is the unique functor M ---> T, where T is the terminal (monoidal) category. Your beta is a right action of this guy on F. In general, a right action of monoidal A --U--> C on mere P --F--> S requires a right action of A on P and a right action of C on S as well as a natural transformation F(p)*U(a) --beta(p,a)--> F(p*a) satisfying the appropriate associativity and unit axioms. In your case S is equipped with the trivial right T-action (x*1=x), and M with its canonical right M-action (a*b=a*b). The axioms are identical. Cheers, Jeff. --- Paul B Levy <P.B.Levy@cs.bham.ac.uk> 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
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