From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3103 Path: news.gmane.org!not-for-mail From: "Stephen Lack" Newsgroups: gmane.science.mathematics.categories Subject: RE: An autonomous category Date: Wed, 15 Mar 2006 11:58:36 +1100 Message-ID: <039A7CE5BC8F554C81732F2505D511720290EF4B@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 1241019096 7261 80.91.229.2 (29 Apr 2009 15:31:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:31:36 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Wed Mar 15 19:18:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:49 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJf8G-00032v-7q for categories-list@mta.ca; Wed, 15 Mar 2006 19:10:20 -0400 X-MimeOLE: Produced By Microsoft Exchange V6.5.7226.0 Content-class: urn:content-classes:message Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 49 Original-Lines: 53 Xref: news.gmane.org gmane.science.mathematics.categories:3103 Archived-At: Dear Marco, This has been considered by Brian Day. He spoke about it in a talk *-autonomous convolution=20 in the Australian Category Seminar on 5 March 1999, You can also transform this via the log/exponential functions to an=20 additive tensor product on the extended (positive and negative) reals. Regards, Steve Lack. -----Original Message----- From: cat-dist@mta.ca on behalf of Marco Grandis Sent: Tue 14/03/2006 12:45 AM To: categories@mta.ca Subject: categories: An autonomous category =20 The Lawvere category of extended positive real numbers has also an autonomous structure, with a multiplicative tensor product (instead of the original additive one). Has this been considered somewhere? To be more explicit: The well-known article of Lawvere on "Metric spaces..." (Rend. Milano 1974, republished in TAC Reprints n. 1) introduced the category of extended positive real numbers, from 0 to oo (infinity included), with arrows x \geq y, equipped with a strict symmetric monoidal closed structure: the tensor product is the sum, the internal hom is truncated difference (with oo - oo =3D 0). Now, the same category can be equipped with a multiplicative tensor product, x.y. Provided we define 0.oo =3D oo (so that tensoring by any element preserves the initial object oo), this is again a strict symmetric monoidal closed structure, with hom(y, z) =3D z/y. Now, the 'undetermined forms' 0/0 and oo/oo are defined to be 0. The new multiplicative structure is even *-autonomous, with involution x* =3D 1/x (and 'nearly' compact). (Note that this choice of values of the undetermined forms comes from privileging the direction x \geq y, which is necessary if we want to view metric spaces, normed categories etc. as enriched categories). Marco Grandis