From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5754 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: bilax monoidal functors Date: Fri, 7 May 2010 10:59:42 -0400 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= 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 1273251074 1822 80.91.229.12 (7 May 2010 16:51:14 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 7 May 2010 16:51:14 +0000 (UTC) To: "Steve Lack" , "Fred E.J. Linton" Original-X-From: categories@mta.ca Fri May 07 18:51:10 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 1OAQlZ-0004dA-8R for gsmc-categories@m.gmane.org; Fri, 07 May 2010 18:51:09 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OAQO5-00058F-Cm for categories-list@mta.ca; Fri, 07 May 2010 13:26:53 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5754 Archived-At: Dear All, In the chapter 3 of their book "Monoidal functor, species and Hopf algebras" http://www.math.tamu.edu/~maguiar/ Aguiar and Mahajan introduces 4 kinds of monoidal functors: 1) strong monoidal 2) lax monoidal 3) colax monoidal 4) bilax monoidal A monoid in a monoidal category C=20 is a lax monoidal functor 1-->C,=20 a comonoid is a colax monoidal functor 1-->C=20 and a bimonoid is a bilax monoidal functor 1-->C. I wonder who first introduced the notion of bilax monoidal functor and when? An example of bilax monoidal functor is the singuler chain complex functor from=20 spaces to chain complexes. The bilax structure is provided by the Eilenberg-MacLane map together with the Alexander-Whitney map. Best, AJ -------- Message d'origine-------- De: categories@mta.ca de la part de Steve Lack Date: jeu. 06/05/2010 19:02 =C0: Fred E.J. Linton; categories Objet : Re: categories: Q. about monoidal functors =20 Dear Fred, Such a T is called a symmetric monoidal functor. Example: let _A_ be Set with the cartesian monoidal structure. Let M be a monoid and let T be the functor Set->Set sending X to MxX (which I'll write as MX). This functor T is monoidal via the map MXMY->MXY = sending (m,x,n,y) to (mn,x,y). It is symmetric monoidal iff M is commutative. Steve Lack. On 6/05/10 4:01 PM, "Fred E.J. Linton" wrote: > Suppose _A_ is a symmetric monoidal category in the sense > of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_ > a monoidal functor. > > What, if anything, is known, where τ: X ⊗ Y --> Y ⊗ = X > is the symmetry structure on the (symmetric) tensor product ⊗, > as to whether > > [T_X,Y: TX ⊗ TY --> T(X ⊗ Y)] > and > [T(τ_X,Y): T(X ⊗ Y) --> T(Y ⊗ X)] > > have the same composition as have > > [τ_TX,TY: TX ⊗ TY --> TY ⊗ TX] > and > [T_Y,X: TY ⊗ TX --> T(Y ⊗ X)] ? > > TIA for any relevant information and/or references thereto. > > Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]