From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5763 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: bilax monoidal functors Date: Sat, 08 May 2010 19:38:14 +1000 Message-ID: References: Reply-To: Steve Lack 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 1273359526 23402 80.91.229.12 (8 May 2010 22:58:46 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 8 May 2010 22:58:46 +0000 (UTC) To: John Baez , categories Original-X-From: categories@mta.ca Sun May 09 00:58:42 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 1OAsyn-0004Oy-O5 for gsmc-categories@m.gmane.org; Sun, 09 May 2010 00:58:41 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OAsSx-0001cL-O5 for categories-list@mta.ca; Sat, 08 May 2010 19:25:47 -0300 Thread-Topic: categories: bilax monoidal functors Thread-Index: Acruki4gIQ1osMuyPkqZUqKGirq/BQ== In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5763 Archived-At: On 8/05/10 4:03 AM, "John Baez" wrote: > Andr=E9 Joyal wrote: >=20 >=20 >> I wonder who first introduced the notion of bilax monoidal functor and >> when? >>=20 >=20 > I believe that Aguiar and Mahajan were the first to formally introduce th= is > concept, though the Alexander-Whitney-Eilenberg-MacLane example has been > around for a long time. This is also my belief. >=20 > On the n-Category Cafe, Kathryn Hess recently wrote: >=20 >> The A-W/E-Z equivalences for the normalized chains functor are a special >> case of the strong deformation retract of chain complexes that was >> constructed by Eilenberg and MacLane in their 1954 Annals paper "On the >> groups H(=BC,n). II". For any commutative ring R, they defined chain >> equivalences between the tensor product of the normalized chains on two >> simplicial R-modules and the normalized chains on their levelwise tensor >> product. >>=20 >> Steve Lack and I observed recently that the normalized chains functor is >> actually even Frobenius monoidal. We then discovered that Aguiar and Mah= ajan >> already had a proof of this fact in their recent monograph. :-) >>=20 > I forget if "Frobenius monoidal" is a precise synonym of "bilax monoidal"= . >=20 No it's not. Frobenius monoidal is to Frobenius algebras as bilax monoidal is to bialgebras. In particular a Frobenius monoidal functor 1-->C is a Frobenius algebra in C; a bilax monoidal functor 1-->C is a bialgebra in C. Steve. > Best, > jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]