From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6012 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor of monads Date: Fri, 30 Jul 2010 23:41:44 +0100 (BST) Message-ID: References: <4C5224A4.4000105@informatik.uni-bremen.de> Reply-To: Tom Leinster NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; format=flowed; charset=US-ASCII X-Trace: dough.gmane.org 1280613422 2112 80.91.229.12 (31 Jul 2010 21:57:02 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 31 Jul 2010 21:57:02 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Jul 31 23:57:00 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OfK38-0002LZ-AX for gsmc-categories@m.gmane.org; Sat, 31 Jul 2010 23:56:58 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46546) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OfK0u-00029v-Am; Sat, 31 Jul 2010 18:54:40 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OfK0j-0001ua-W0 for categories-list@mlist.mta.ca; Sat, 31 Jul 2010 18:54:30 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6012 Archived-At: 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/ ]