From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6516 Path: news.gmane.org!not-for-mail From: N.Bowler@dpmms.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: RE: categories with several compositions? Date: 03 Feb 2011 11:56:38 +0000 Message-ID: References: Reply-To: N.Bowler@dpmms.cam.ac.uk NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 X-Trace: dough.gmane.org 1296833058 3749 80.91.229.12 (4 Feb 2011 15:24:18 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 4 Feb 2011 15:24:18 +0000 (UTC) Cc: "'categories@mta.ca'" To: John Stell Original-X-From: majordomo@mlist.mta.ca Fri Feb 04 16:24:14 2011 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 1PlNW9-0008Uk-ID for gsmc-categories@m.gmane.org; Fri, 04 Feb 2011 16:24:13 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:34163) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PlNVz-0006up-Tl; Fri, 04 Feb 2011 11:24:03 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PlNVu-0008Dd-F1 for categories-list@mlist.mta.ca; Fri, 04 Feb 2011 11:23:58 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6516 Archived-At: I've just noticed there was a bit more to your question: >More generally, a kind of category with several compositions: >for each object y there is a set Dy and instead of the usual > >C(x,y) x C(y,z) -> C(x,z) > >we have Dy -> [C(x,y) x C(y,z), C(x,z)] > >So you have a family of compositions at each object which "associate with >each other" in the manner of the above equation, and where there is >a single identity for each object. I assume you would now want an identity at each object for each composition. Then exactly the same argument as in my last email shows that structures like this can be analysed in terms of categories C with a designated family of (assignments to each object a of C of an invertible endomorphism of a). Nathan PS There's a small typo in my last email. Replace `s_i' by `e_i'. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]