bilax monoidal functors

["Joyal, André" <joyal.andre@uqam.ca>]

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

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