Re: Q. about monoidal functors

Re: Q. about monoidal functors

[Fred E.J. Linton]

Thanks, Steve,

> Such a T is called a symmetric monoidal functor.

Thanks for helping dispel my illusion that all monoidal
functors might necessarily be thus symmetric :-) :
 
> Example: let _A_ be Set with the cartesian monoidal structure. Let
> M be a monoid and let T be the functor Set->Set sending X to MxX (which
> I'll write as MX). This functor T is monoidal via the map MXMY->MXY sending
> (m,x,n,y) to (mn,x,y). It is symmetric monoidal iff M is commutative.

Cheers, -- Fred




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Re: Q. about monoidal functors

[Toby Bartels]

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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Re: Q about_monoidal_functors?

[Andre Joyal]

Dear Toby,

Looping and delooping operations can be applied to spaces
and to maps between spaces. 
We should use a similar terminology for spaces and maps.
For example:

E-n space <--> E-n map


Also for (higher) categories and functors. 

monoidal category <---> monoidal functor
braided monoidal category <----> braided monoidal functor
2-braided monoidal category <--> 2-braided monoidal functor
3-braided monoidal category <--> 3-braided monoidal functor
......
......
......

symmetric monoidal category <--> symmetric monoidal functor


A (n+1)-braided monoidal n-category is symmetric by 
the stabilisation hypothesis. 
I believe that a (n+1)-braided monoidal functor 
between (n+1)-braided monoidal n-categories is symmetric. 
Is this part of the official stabilisation hypothesis?

Best,
André 


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Re: Q about_monoidal_functors?

[Toby Bartels]

Andre Joyal wrote in part:

>We should use a similar terminology for spaces and maps.
>E-n space <--> E-n map
>Also for (higher) categories and functors.
>monoidal category <---> monoidal functor
>braided monoidal category <----> braided monoidal functor
>2-braided monoidal category <--> 2-braided monoidal functor
>3-braided monoidal category <--> 3-braided monoidal functor
>......
>symmetric monoidal category <--> symmetric monoidal functor

I agree, one should say "symmetric monoidal functor";
if nothing else, that indicates that the source and target
are symmetric (not merely braided) monoidal categories.
I only put "BMF" in my table to show a particular pattern.

Depending on how you write down the definitions,
that a braided monoidal functor between symmetric monoidal categories
is the same thing as a symmetric monoidal functor between them
is either an utter triviality or a deep and interesting theorem;
but in either case, we need the words to state it.

(I do agree with John about preferring "k-tuply monoidal",
  but I'll let him make that argument.)

>A (n+1)-braided monoidal n-category is symmetric by
>the stabilisation hypothesis.
>I believe that a (n+1)-braided monoidal functor
>between (n+1)-braided monoidal n-categories is symmetric.

I think that you mean to say (which is even stronger)
that an n-braided monoidal functor between SM n-categories is symmetric.
More generally, a k-braided monoidal l-transfor between SM n-categories
is symmetric as long as k + l is greater than or equal to n.
(A 0-transfor is a functor, a 1-transfor is a natural transformation, etc.
  This numbering is due to Sjoerd Crans; feel free to argue that it's off.)

More generally yet, a k-braided monoidal l-transfor
between m-braided monoidal n-categories is m-braided,
as long as k + l >= n, regardless of the value of m
(although we need m >= k for the antecedent to make sense).

>Is this part of the official stabilisation hypothesis?

I don't know what's official, but I'll claim the conjecture above as mine
if nobody else has written it down yet.  (^_^)


--Toby


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wrong axioms

[Andre Joyal]

Dear John,

I want you and everyone know that the definition of a braiding in your notes 

* John Baez, Some definitions everyone should know.
    http://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf


is wrong. The notes were recently publicised by Toby.

But the definition given in the nLab 

http://ncatlab.org/nlab/show/braided+monoidal+category

is correct.


Best, 
André


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