From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3384 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Linear--structure or property? Date: Thu, 10 Aug 2006 16:14:46 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019272 8530 80.91.229.2 (29 Apr 2009 15:34:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:32 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Thu Aug 10 20:37:10 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 10 Aug 2006 20:37:10 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GBJzo-0007Oz-OI for categories-list@mta.ca; Thu, 10 Aug 2006 20:31:24 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 17 Original-Lines: 16 Xref: news.gmane.org gmane.science.mathematics.categories:3384 Archived-At: 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.