From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10881 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: ptj@maths.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Re: Non-cartesian closedness of Met Date: 18 Dec 2022 13:04:34 +0000 Message-ID: Reply-To: ptj@maths.cam.ac.uk Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="28981"; mail-complaints-to="usenet@ciao.gmane.io" Cc: Categories mailing list To: =?ISO-8859-1?Q?Jir=ED_Ad=E1mek?= Original-X-From: majordomo@rr.mta.ca Tue Dec 20 03:21:28 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 1p7SFz-0007KF-6p for gsmc-categories@m.gmane-mx.org; Tue, 20 Dec 2022 03:21:27 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:37500) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1p7SFD-0006kZ-5C; Mon, 19 Dec 2022 22:20:39 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1p7SEY-000150-TN for categories-list@rr.mta.ca; Mon, 19 Dec 2022 22:19:58 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10881 Archived-At: Dear Jirka, That's an interesting idea, but I'm having difficulty making it work in practice. It seems to me that the most you can get from considering morphisms from 2 is that the metric on [X,Y] must satisfy d(f,g) \geq sup_x d(fx,gx) -- to get the reverse inequality you would=20 need to impose the L_1 metric on the product 2 x X. And that inequality is the wrong way round for showing that the transpose of addition on R is nonexpansive. Best regards, Peter On Dec 16 2022, Jir=ED Ad=E1mek wrote: >Hi Peter, > >If Met is a CCC, then [X,Y] has as elements all morphisms from X to Y >(use the adjoint transposes of morphisms from 1 to [X,Y]). And the=20 >distance of morphisms f,g is sup_x d(fx,gx) (use the adjoint transposes >of morphisms from two-element spaces to [X,Y]). > >However, addition of real numbers is not nonexpansive from R x R to R,=20 >although its curred form from R to [R,R] is. This is a contradiciotn. > >Best regards, >Jiri > >On Fri, 16 Dec 2022, ptj@maths.cam.ac.uk wrote: > >> 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/ ]