From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3389 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Linear--structure or property? Date: Fri, 11 Aug 2006 11:12:38 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019275 8540 80.91.229.2 (29 Apr 2009 15:34:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:35 +0000 (UTC) To: "Categories list" Original-X-From: rrosebru@mta.ca Fri Aug 11 09:50:38 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 11 Aug 2006 09:50:38 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GBWNc-0006xJ-W2 for categories-list@mta.ca; Fri, 11 Aug 2006 09:44:49 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 22 Original-Lines: 53 Xref: news.gmane.org gmane.science.mathematics.categories:3389 Archived-At: Dear Michael, I have a trivial comment to the first paragraph of your message. You ask: "...Could you have two (semi)ring structures on the same set with the same associative multiplication?" Take any (semi)ring R with a multiplicative automorphism f that is not an additive automorphism, and transport the structure along f. For instance, both in the semiring of natural numbers and in the ring of integers, any non-identity permutation of (positive) prime numbers determines such an f. An example for students: take, say, f(2) = 3, f(3) = 2, and f(p) = p for all primes p different from 2 and 3. Then, denoting the new addition by *, we calculate (usung the fact that f coincides with its inverse) 1*1 = f(f(1)+f(1)) = f(2+2) = f(2x2) = f(2)xf(2) = 3x3 = 9. Best regards, George ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Thursday, August 10, 2006 10:14 PM Subject: categories: Linear--structure or property? > Bill Lawvere uses "linear" for a category enriched over commutative > semigroups. Obviously, if the category has finite products, this is a > property. What about in the absence of finite products (or sums)? Could > you have two (semi)ring structures on the same set with the same > associative multiplication? > > Robin Houston's startling (to me, anyway) proof that a compact > *-autonomous category with finite products is linear starts by proving > that 0 = 1. Suppose the category has only binary products? Well, I have > an example of one that is not linear: Lawvere's category that is the > ordered set of real numbers has a compact *-autonomous structure. > Tensor is + and internal hom is -. Product is inf and sum is sup, but > there are no initial or terminal objects and the category is not linear. > > > >