Q. about monoidal functors

Q. about monoidal functors

[Fred E.J. Linton]

Suppose _A_ is a symmetric monoidal category in the sense
of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_
a monoidal functor.

What, if anything, is known, where τ: X ⊗ Y --> Y ⊗ X
is the symmetry structure on the (symmetric) tensor product ⊗, 
as to whether

[T_X,Y: TX ⊗ TY --> T(X ⊗ Y)] 
and 
[T(τ_X,Y): T(X ⊗ Y) --> T(Y ⊗ X)]

have the same composition as have

[τ_TX,TY: TX ⊗ TY --> TY ⊗ TX]
and
[T_Y,X: TY ⊗ TX --> T(Y ⊗ X)] ?

TIA for any relevant information and/or references thereto.

Cheers, -- Fred





[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Q. about monoidal functors

[Steve Lack]

Dear Fred,

Such a T is called a symmetric monoidal functor.

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.

Steve Lack.


On 6/05/10 4:01 PM, "Fred E.J. Linton" <fejlinton@usa.net> wrote:

> Suppose _A_ is a symmetric monoidal category in the sense
> of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_
> a monoidal functor.
>
> What, if anything, is known, where &tau;: X &otimes; Y --> Y &otimes; X
> is the symmetry structure on the (symmetric) tensor product &otimes;,
> as to whether
>
> [T_X,Y: TX &otimes; TY --> T(X &otimes; Y)]
> and
> [T(&tau;_X,Y): T(X &otimes; Y) --> T(Y &otimes; X)]
>
> have the same composition as have
>
> [&tau;_TX,TY: TX &otimes; TY --> TY &otimes; TX]
> and
> [T_Y,X: TY &otimes; TX --> T(Y &otimes; X)] ?
>
> TIA for any relevant information and/or references thereto.
>
> Cheers, -- Fred
>
>
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

bilax monoidal functors

[Joyal, André]

Dear All,

In the chapter 3 of their book

"Monoidal functor, species and Hopf algebras"

http://www.math.tamu.edu/~maguiar/

Aguiar and Mahajan introduces 4 kinds of monoidal functors:

1) strong monoidal
2) lax monoidal
3) colax monoidal
4) bilax monoidal

A monoid in a monoidal category C 
is a lax monoidal functor 1-->C, 
a comonoid is a colax monoidal functor 1-->C 
and a bimonoid is a bilax monoidal functor 1-->C.

I wonder who first introduced the notion
of bilax monoidal functor and when?

An example of bilax monoidal functor is
the singuler chain complex functor from 
spaces to chain complexes. The bilax structure
is provided by the Eilenberg-MacLane map
together with the Alexander-Whitney map.

Best,
AJ



-------- Message d'origine--------
De: categories@mta.ca de la part de Steve Lack
Date: jeu. 06/05/2010 19:02
À: Fred E.J. Linton; categories
Objet : Re: categories: Q. about monoidal functors
 
Dear Fred,

Such a T is called a symmetric monoidal functor.

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.

Steve Lack.


On 6/05/10 4:01 PM, "Fred E.J. Linton" <fejlinton@usa.net> wrote:

> Suppose _A_ is a symmetric monoidal category in the sense
> of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_
> a monoidal functor.
>
> What, if anything, is known, where &tau;: X &otimes; Y --> Y &otimes; X
> is the symmetry structure on the (symmetric) tensor product &otimes;,
> as to whether
>
> [T_X,Y: TX &otimes; TY --> T(X &otimes; Y)]
> and
> [T(&tau;_X,Y): T(X &otimes; Y) --> T(Y &otimes; X)]
>
> have the same composition as have
>
> [&tau;_TX,TY: TX &otimes; TY --> TY &otimes; TX]
> and
> [T_Y,X: TY &otimes; TX --> T(Y &otimes; X)] ?
>
> TIA for any relevant information and/or references thereto.
>
> Cheers, -- Fred

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]