Re: Q. about monoidal functors

[Toby Bartels <toby@ugcs.caltech.edu>]

Fred E.J. Linton wrote in part:

>Steve Lack wrote:

>>Such a T is called a symmetric monoidal functor.

>Thanks for helping dispel my illusion that all monoidal
>functors might necessarily be thus symmetric :-) :

Something like this is true, however.

First, every monoidal natural transformation is symmetric monoidal
(assuming that it goes between symmetric monoidal functors at all).
Also, there is the concept of braided monoidal categories that lies
between monoidal categories and symmetric monoidal categories.
And every braided monoidal functor is symmetric monoidal
(assuming that it goes between symmetric monoidal categories at all).

Each of these facts is trivial by itself; for example,
the definition of symmetric monoidal functor that you wrote down
makes sense for a functor between braided monoidal categories;
it is simply the definition of braided monoidal functor,
and there is nothing more to add when the braiding is symmetric.

But the entire pattern is interesting:

  PC --  PF -- PNT -- ENT
  MC --  MF -- MNT -- ENT
BMC -- BMF -- MNT -- ENT
SMC -- BMF -- MNT -- ENT
SMC -- BMF -- MNT -- ENT
SMC -- BMF -- MNT -- ENT
(etc)

(To fit this all on the screen, I have used initialisms:
"Categories", "Functors", "Natural transformations", "Equality of",
"Pointed", "Monoidal", "Braided", "Symmetric".)

The thing to notice is that each column stabilises
one row earlier than the column before it.
The columns stabilise because there is nothing more to write down.

* John Baez, Some definitions everyone should know.
   http://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf
(This discusses strong monoidal functors between weak monoidal categories,
  but it is easy enough to generalise to lax monoidal functors
  or to specialise to strict monoidal categories.)

It's possible that the columns stabilise only through our ignorance
(as once we were ignorant that BMC were there between MC and SMC).
However, there is a general theory of k-tuply monoidal n-categories
which confirms the pattern, although some of that is still conjecture.

* nLab, k-tuply monoidal n-categories
   http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category


--Toby


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