Re: bilax monoidal functors

[Steve Lack <s.lack@uws.edu.au>]

On 8/05/10 4:03 AM, "John Baez" <john.c.baez@gmail.com> wrote:

> André Joyal wrote:
> 
> 
>> I wonder who first introduced the notion of bilax monoidal functor and
>> when?
>> 
> 
> I believe that Aguiar and Mahajan were the first to formally introduce this
> concept, though the Alexander-Whitney-Eilenberg-MacLane example has been
> around for a long time.

This is also my belief.

> 
> On the n-Category Cafe, Kathryn Hess recently wrote:
> 
>> The A-W/E-Z equivalences for the normalized chains functor are a special
>> case of the strong deformation retract of chain complexes that was
>> constructed by Eilenberg and MacLane in their 1954 Annals paper "On the
>> groups H(¼,n). II". For any commutative ring R, they defined chain
>> equivalences between the tensor product of the normalized chains on two
>> simplicial R-modules and the normalized chains on their levelwise tensor
>> product.
>> 
>> Steve Lack and I observed recently that the normalized chains functor is
>> actually even Frobenius monoidal. We then discovered that Aguiar and Mahajan
>> already had a proof of this fact in their recent monograph. :-)
>> 
> I forget if "Frobenius monoidal" is a precise synonym of "bilax monoidal".
> 

No it's not. Frobenius monoidal is to Frobenius algebras as bilax monoidal
is to bialgebras.

In particular a Frobenius monoidal functor 1-->C is a Frobenius algebra in
C; a bilax monoidal functor 1-->C is a bialgebra in C.

Steve.


> Best,
> jb



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