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