bilax monoidal functors

[David Yetter <dyetter@math.ksu.edu>]

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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