From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6021 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor of monads Date: Sun, 01 Aug 2010 20:49:25 +0100 Message-ID: References: <4C5224A4.4000105@informatik.uni-bremen.de> Reply-To: Ronnie Brown NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1280752361 6860 80.91.229.12 (2 Aug 2010 12:32:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 2 Aug 2010 12:32:41 +0000 (UTC) Cc: categories@mta.ca To: Tom Leinster Original-X-From: majordomo@mlist.mta.ca Mon Aug 02 14:32:40 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OfuC5-0000Ih-Lu for gsmc-categories@m.gmane.org; Mon, 02 Aug 2010 14:32:38 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44595) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OfuBE-0003k3-AA; Mon, 02 Aug 2010 09:31:44 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OfuBA-0006nE-Rg for categories-list@mlist.mta.ca; Mon, 02 Aug 2010 09:31:40 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6021 Archived-At: Looking at the two ways of evaluating (say with . vertically and * horizontally) 1 e e 1 is what gives 1=e but it seems to need two sided units. Ronnie Tom Leinster wrote: > On Fri, 30 Jul 2010, Prof. Peter Johnstone wrote: > >> On Fri, 30 Jul 2010, Sergey Goncharov wrote: >> >>> This looks fine! But I guess, Eckmann-Hilton argument does not apply >>> to your previous example because it presupposes that the monoidal >>> structures share the unit, which was not the case there, was it? >>> >> Yes, it was: the fact that the unit for each semilattice structure is >> a homomorphism for the other forces them to be the same. > > In any case, it doesn't matter: the Eckmann-Hilton argument *doesn't* > presuppose that the monoid structures share the unit. > > Here are the weakest hypotheses I know for the elementary Eckmann-Hilton > argument: > > Let A be a set. > Let . be a binary operation on A with two-sided unit 1. > Let * be a binary operation on A with two-sided unit e. > Suppose that > > (a * b) . (a' * b') = (a . a') * (b . b') > > for all a, b, a', b' in A. > Then . = *, 1 = e, and (A, ., 1) is a commutative monoid. > > So equality of the units and associativity, as well as the more familiar > stuff, come for free. (I bet you can weaken "two-sided", too.) > > Tom > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]