From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3388 Path: news.gmane.org!not-for-mail From: "Stephen Lack" Newsgroups: gmane.science.mathematics.categories Subject: RE: Linear--structure or property? Date: Fri, 11 Aug 2006 20:49:15 +1000 Message-ID: <039A7CE5BC8F554C81732F2505D511720290F225@BONHAM.AD.UWS.EDU.AU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019274 8539 80.91.229.2 (29 Apr 2009 15:34:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:34 +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 1GBWPf-00075j-RR for categories-list@mta.ca; Fri, 11 Aug 2006 09:46:56 -0300 Content-class: urn:content-classes:message Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 21 Original-Lines: 42 Xref: news.gmane.org gmane.science.mathematics.categories:3388 Archived-At: It's a structure. Consider the following category C. Two objects x and y, with hom-categories C(x,x)=3DC(y,y)=3D{0,1} C(y,x)=3D{0} C(x,y)=3DM with composition defined so that each 1 is an=20 identity morphism and each 0 a zero morphism, and with M an arbitrary set. Any commutative=20 monoid structure on M makes C into a linear category.=20 Steve. -----Original Message----- From: cat-dist@mta.ca on behalf of Michael Barr Sent: Fri 8/11/2006 6:14 AM To: Categories list Subject: categories: Linear--structure or property? =20 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 =3D 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.