From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6400 Path: news.gmane.org!not-for-mail From: "Ellis D. Cooper" Newsgroups: gmane.science.mathematics.categories Subject: Severe Strict Monoidal Category Naivete Date: Fri, 03 Dec 2010 10:51:20 -0500 Message-ID: References: Reply-To: "Ellis D. Cooper" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Trace: dough.gmane.org 1291427289 5070 80.91.229.12 (4 Dec 2010 01:48:09 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 4 Dec 2010 01:48:09 +0000 (UTC) Cc: categories@mta.ca To: Steve Lack Original-X-From: majordomo@mlist.mta.ca Sat Dec 04 02:48:05 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1POhEL-0000mk-F0 for gsmc-categories@m.gmane.org; Sat, 04 Dec 2010 02:48:05 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:51830) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1POhEB-00063f-L4; Fri, 03 Dec 2010 21:47:55 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1POhE7-0002RO-S1 for categories-list@mlist.mta.ca; Fri, 03 Dec 2010 21:47:51 -0400 In-Reply-To: <8C65074A-894A-4F7A-B47D-9D8411A9CFC3@mq.edu.au> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6400 Archived-At: Dear Steve, At 09:54 PM 12/2/2010, you wrote: If I understand correctly, you have arrows f:X->Y and g:Y->Z and you are comparing the tensor products f@g:X@Y->Y@Z and g@f:Y@X->Z@Y. They have different domain and codomain, so cannot be equal. I was thinking more about f:X->Y and g:Y->Z and tensoring f with the identity morphism of Y to get f@1_Y:X@Y->Y@Y, and also tensoring 1_Y with g to get 1_Y@g:Y@Y->Y@Z. So I get the composition f@1_Y followed by 1_Y@g is a morphism from X@Y->Y@Z, and you made me realize that f followed by g as a morphism X->Y cannot possibly equal f@1_Y followed by 1_Y@g. Then again, if the ambient strict monoidal category is symmetric, so that the latter composition is a morphism X@Y->Z@Y, then to my mind somehow this is pretty much the same as the composition X->Z of f followed by g, basically because only the identity morphism of Y is involved. The context of my inquiry is chemical reaction, as suggested by John Baez a while ago. That is, if f and g are chemical reactions that transform X to Y and Y to Z, respectively, then the net effect is just transformation of X to Y, since the Y produced by f is completely consumed by g. Bottom line: I would like a correct way to say that tensoring f with an identity morphism is somehow no different from f. Ellis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]