From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5772 Path: news.gmane.org!not-for-mail From: Andre Joyal Newsgroups: gmane.science.mathematics.categories Subject: Re: bilax_monoidal_functors?= Date: Mon, 10 May 2010 12:14:40 -0400 Message-ID: References: Reply-To: Andre Joyal NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1273539849 18048 80.91.229.12 (11 May 2010 01:04:09 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 11 May 2010 01:04:09 +0000 (UTC) To: "David Yetter" , "Categories" Original-X-From: categories@mta.ca Tue May 11 03:04:07 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OBdtF-00018f-Oh for gsmc-categories@m.gmane.org; Tue, 11 May 2010 03:04:05 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OBdFV-0002Sl-M0 for categories-list@mta.ca; Mon, 10 May 2010 21:23:01 -0300 Thread-Topic: categories: bilax monoidal functors Thread-Index: AcrvAcW/9fcRBrIORvulTXWGkcQSsABWL6f2 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5772 Archived-At: 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=20 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 (=3Doplax) 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=20 is not defined in their book. Any idea? Best regards, Andr=E9 -------- Message d'origine-------- De: categories@mta.ca de la part de David Yetter Date: ven. 07/05/2010 21:05 =C0: Categories Objet : categories: bilax monoidal functors =20 John Baez could not recall whether bilax and Frobenius monoidal functors = =3D are the same. The answer is no, in the usage I'd been familiar with, bilax meant =3D simply equipped with both lax and oplax structures, while a Frobenius = =3D monoidal functor satisfies additional coherence relation which =3D generalize the relations between the multiplication and comultiplication = =3D in a Frobenius algebra. A bilax monoidal functor from the one-object monoidal category to VECT = =3D would be a vector-space with both an algebra and a coalgebra structure = =3D on it (no coherence relations relating them), while a Frobenius monoidal = =3D functor would be a Frobenius algebra. =3D20 Aguiar (with good reason), on the other hand, reserves bilax for =3D functors equipped with coherence relations generalizing the relations = =3D between the operations and cooperations in a bialgebra, so that a bilax = =3D functor from the one-object monoidal category to VECT would be a =3D bialgebra. This notion, however, only makes sense in the presence of = =3D braidings on the source and target. I think Aguiar's usage should prevail, though we also need a name for = =3D functors between general monoidal categories which are simultaneously = =3D lax and oplax. Best Thoughts, David Yetter=3D [For admin and other information see: http://www.mta.ca/~cat-dist/ ]