From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3110 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: preprint: Categories, norms and weights Date: Wed, 15 Mar 2006 19:18:16 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019100 7290 80.91.229.2 (29 Apr 2009 15:31:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:31:40 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Mar 15 19:19:10 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:19:10 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfGB-0003d8-Uz for categories-list@mta.ca; Wed, 15 Mar 2006 19:18:31 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 56 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:3110 Archived-At: The following preprint is available: Marco Grandis Categories, norms and weights Dip. Mat. Univ. Genova, Preprint 538 (2006), 14 p. http://www.dima.unige.it/~grandis/wCat.pdf http://www.dima.unige.it/~grandis/wCat.ps Abstract. The well-known Lawvere category R of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But R has another such structure, given by multiplication, which is *-autonomous and a CL-algebra (linked with classical linear logic). Normed sets, with a norm in R, inherit thus two symmetric monoidal closed structures, and categories enriched on one of them have a 'subadditive' or 'submultiplicative' norm, respectively. Typically, the first case occurs when the norm expresses a cost, the second with Lipschitz norms. This paper is a preparation for a sequel, devoted to 'weighted algebraic topology', an enrichment of directed algebraic topology. The structure of R, and its extension to the complex projective line, might be a first step in abstracting a notion of algebra of weights, linked with physical measures.