From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5761 Path: news.gmane.org!not-for-mail From: David Yetter Newsgroups: gmane.science.mathematics.categories Subject: bilax monoidal functors Date: Fri, 7 May 2010 20:05:54 -0500 Message-ID: Reply-To: David Yetter NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1078) Content-Transfer-Encoding: quoted-printableContent-Type: text/plain;charset=us-ascii X-Trace: dough.gmane.org 1273359418 23169 80.91.229.12 (8 May 2010 22:56:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 8 May 2010 22:56:58 +0000 (UTC) To: Categories Original-X-From: categories@mta.ca Sun May 09 00:56:57 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 1OAsx4-0003sQ-1r for gsmc-categories@m.gmane.org; Sun, 09 May 2010 00:56:54 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OAsOI-0001WM-Ro for categories-list@mta.ca; Sat, 08 May 2010 19:20:58 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5761 Archived-At: 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/ ]