From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10877 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: ptj@maths.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Non-cartesian closedness of Met Date: 16 Dec 2022 16:41:00 +0000 Message-ID: Reply-To: ptj@maths.cam.ac.uk Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="19928"; mail-complaints-to="usenet@ciao.gmane.io" To: Categories mailing list Original-X-From: majordomo@rr.mta.ca Fri Dec 16 21:34:38 2022 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.75]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1p6HPi-0004vM-4s for gsmc-categories@m.gmane-mx.org; Fri, 16 Dec 2022 21:34:38 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:37278) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1p6HPG-0006PR-IH; Fri, 16 Dec 2022 16:34:10 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1p6HOO-0005FO-RF for categories-list@rr.mta.ca; Fri, 16 Dec 2022 16:33:16 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10877 Archived-At: Let Met denote the category of metric spaces and nonexpansive maps. It's well known that if we equip the product of two metric spaces with the L_{\infty} metric (the max of the distances in the two coordinates), we get categorical products in Met; alternatively, if we impose the L_1 metric on the product (the sum of the two coordinate distances), we get a monoidal closed structure, at least if we weaken the usual definition of a metric by allowing metrics to take the value \infty. It's intuitively obvious that the cartesian monoidal structure on Met can't be closed. But I've never (until I wrote one down today!) seen a formal proof of this; does anyone know if it exists anywhere in the literature? My proof is not particularly elegant: it amounts to showing that a particular coequalizer in Met is not preserved by a functor of the form (-) x Y. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]