From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5759 Path: news.gmane.org!not-for-mail From: Andre Joyal Newsgroups: gmane.science.mathematics.categories Subject: Re: bilax monoidal functors Date: Fri, 7 May 2010 22:23:58 -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 1273359417 23163 80.91.229.12 (8 May 2010 22:56:57 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 8 May 2010 22:56:57 +0000 (UTC) To: "John Baez" , "categories" Original-X-From: categories@mta.ca Sun May 09 00:56:56 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 1OAsx1-0003qh-37 for gsmc-categories@m.gmane.org; Sun, 09 May 2010 00:56:51 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OAsPf-0001X6-Q2 for categories-list@mta.ca; Sat, 08 May 2010 19:22:23 -0300 Thread-Topic: categories: bilax monoidal functors Thread-Index: AcruP7uOyWrNLPnfQFeHtYqMAu4WewAFcg/F Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5759 Archived-At: Dear John, I am using the following terminology for higher braided monoidal (higher) categories: Monoidal< braided < 2-braided <....... I wonder who first introduced the notion of bilax monoidal functor and > when? > I believe that Aguiar and Mahajan were the first to formally introduce = this concept, though the Alexander-Whitney-Eilenberg-MacLane example has been around for a long time. On the n-Category Cafe, Kathryn Hess recently wrote: > 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(?,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. > > Steve Lack and I observed recently that the normalized chains functor = is > actually even Frobenius monoidal. We then discovered that Aguiar and = Mahajan > already had a proof of this fact in their recent monograph. :-) > I forget if "Frobenius monoidal" is a precise synonym of "bilax = monoidal". Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]