From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10879 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Dirk Hofmann Newsgroups: gmane.science.mathematics.categories Subject: Re: Non-cartesian closedness of Met Date: Sat, 17 Dec 2022 09:20:17 +0000 Message-ID: Reply-To: Dirk Hofmann Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8"; format=flowed Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="38149"; mail-complaints-to="usenet@ciao.gmane.io" Cc: Maria Manuel Clementino , Categories mailing list To: Original-X-From: majordomo@rr.mta.ca Sat Dec 17 23:03:45 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 1p6fHU-0009gp-H9 for gsmc-categories@m.gmane-mx.org; Sat, 17 Dec 2022 23:03:44 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:37410) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1p6fH6-0004Fz-HU; Sat, 17 Dec 2022 18:03:20 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1p6fGU-0000ne-Fc for categories-list@rr.mta.ca; Sat, 17 Dec 2022 18:02:42 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10879 Archived-At: Dear Peter, we don't have such an example in the paper. Our argument used the=20 cartesian closed category of "metric spaces without triangular=20 inequality", and showed that, for a metric space X, all=20 exponential Y^X (Y metric) taken in that category satisfy the=20 triangular inequality if and only if X satisfies that condition. Best regards Dirk On 16 December 2022 at 22:05 GMT+0000, ptj@maths.cam.ac.uk wrote=20 ... > Dear Dirk, dear Maria Manuel, > > That's very interesting, and I should have remembered it. But=20 > did > your argument come up with an explicit example of a colimit in=20 > Met > not preserved by a functor of the form (-) x Y ? > > I suppose I'll have to go and read your paper! > > Best regards, > Peter > > On Dec 16 2022, Dirk Hofmann wrote: > >> >>Dear Peter, >> >>in our paper >> >> - Clementino, M. M., & Hofmann, D. (2006). Exponentiation in >> $V$-categories. Topology and its Applications, 153(16),=20 >> 3113=E2=80=933128. >> >> we give a characterisation of exponentiable metric spaces. The=20 >> result >> essentially states that a metric space (in the sense of=20 >> Lawvere) is >> exponentiable if and only if "there is always a point in the=20 >> middle", >> that is, whenever d(x,z)=3Du+v, then there is a point y with=20 >> d(x,y)=E2=89=A4u+=CE=B5 >> and d(y,z)=E2=89=A4v+=CE=B5. A finite metric space with a non-trivial=20 >> distance >> cannot be exponentiable. >> >>Best regards >>Dirk >> >> On 16 December 2022 at 16:41 GMT+0000, =20 >> wrote ... >>> Let Met denote the category of metric spaces and nonexpansive=20 >>> maps. >>> It's well known that if we equip the product of two metric=20 >>> spaces >>> with the L_{\infty} metric (the max of the distances in the=20 >>> two >>> coordinates), we get categorical products in Met;=20 >>> 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=20 >>> least >>> if we weaken the usual definition of a metric by allowing=20 >>> metrics to >>> take the value \infty. >>> >>> It's intuitively obvious that the cartesian monoidal structure=20 >>> on Met >>> can't be closed. But I've never (until I wrote one down=20 >>> today!) seen >>> a formal proof of this; does anyone know if it exists anywhere=20 >>> in the >>> literature? My proof is not particularly elegant: it amounts=20 >>> to >>> showing >>> that a particular coequalizer in Met is not preserved by a=20 >>> functor of >>> the form (-) x Y. >>> >>> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]