Re: bilax_monoidal_functors?=

[Andre Joyal <joyal.andre@uqam.ca>]

Dear David

Thanks for clarifying the notion of Frobenius functor.

In the chapter 4 of the latest version of their book

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

Aguiar and Mahajan introduce a notion of P-monoidal functor 
for P is an operad.

If P is the Ass operad (whose models are monoids).
then a P-monoidal functor is a lax monoidal functor,
and if P is the Com operad (whose models are commutative monoids),
then a  P-monoidal functor is a symmetric lax monoidal functor.
Their examples include a notion of Lie-monoidal functor (in the enriched case).
Dually, they introduce a notion of P-comonoidal functor
with the examples of colax (=oplax) monoidal functors
and of symmetric oplax monoidal functors.

But a bilax monoidal functor is not a P-monoidal functor
in the sense of Aguiar and Mahajan because the notion of bialgebra
is defined by a PROP, not by an operad. Similarly a Frobenius monoidal functor
is not a P-monoidal functor because the notion of Frobenius algebra
is defined by a PROP, not by an operad.


The notion of P-monoidal functor for P a PROP 
is not defined in their book.

Any idea?


Best regards,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de David Yetter
Date: ven. 07/05/2010 21:05
À: Categories
Objet : categories: bilax monoidal functors
 
John Baez could not recall whether bilax and Frobenius monoidal functors =
are the same.

The answer is no, in the usage I'd been familiar with,  bilax meant =
simply equipped with both lax and oplax structures, while a Frobenius =
monoidal functor satisfies  additional coherence relation which =
generalize the relations between the multiplication and comultiplication =
in a Frobenius algebra.

A bilax monoidal functor from the one-object monoidal category to VECT =
would be a vector-space with both an algebra and a coalgebra structure =
on it (no coherence relations relating them), while a Frobenius monoidal =
functor would be a Frobenius algebra. =20

Aguiar (with good reason), on the other hand, reserves bilax for =
functors equipped with coherence relations generalizing the relations =
between the operations and cooperations in a bialgebra, so that a bilax =
functor from the one-object monoidal category to VECT would be a =
bialgebra.  This notion, however, only makes sense in the presence of =
braidings on the source and target.

I think Aguiar's usage should prevail, though we also need a name for =
functors between general monoidal categories which are simultaneously =
lax and oplax.

Best Thoughts,
David Yetter=


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