From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7830 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Non-triviality of *-autonomous categories Date: Sun, 4 Aug 2013 09:54:04 -0400 (EDT) Message-ID: References: <77BE9A54-A589-408D-89B0-9C99D20B7F45@gmail.com> Reply-To: Michael Barr NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1375659540 13631 80.91.229.3 (4 Aug 2013 23:39:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 4 Aug 2013 23:39:00 +0000 (UTC) Cc: categories@mta.ca To: "Harley D. Eades III" Original-X-From: majordomo@mlist.mta.ca Mon Aug 05 01:39:02 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1V67t6-0004mR-JR for gsmc-categories@m.gmane.org; Mon, 05 Aug 2013 01:39:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58619) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1V67rN-0001yJ-Lq; Sun, 04 Aug 2013 20:37:13 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1V67rO-0008Q0-JN for categories-list@mlist.mta.ca; Sun, 04 Aug 2013 20:37:14 -0300 In-Reply-To: <77BE9A54-A589-408D-89B0-9C99D20B7F45@gmail.com> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7830 Archived-At: In the original definition, the fact that A --> A** is an isomorphism was part of the definition. If that is not what you're asking, then I don't understand the question. Of course you have to prove that for Chu categories, but that was relatively easy. On Sat, 3 Aug 2013, Harley D. Eades III wrote: > Hi, everyone. > > I am having trouble finding a reference. I thought perhaps someone here > might know. > > It is well known that adding the isomorphism: > d : A -> (A => 1) => 1 > to a bicartisan closed category degenerates to a preorder. > > In *-autonomous categories we have such an isomorphism, but > is non-trivial. Where can I find a proof of this? I would like to > reference it. > > I think one could proof this using the category of coherence spaces and linear maps > as a concrete *-autonomous category. See for example: > > [1] R. a. g. Seely. Linear logic, *-autonomous categories and cofree coalgebras. In Computer Science Logic, 1989. > > Any references anyone might have would be great. > > Thanks, > .\ Harley > > The modern conservative is engaged in one of man's oldest exercises in moral philosophy--the search for a superior moral justification for selfishness. --J.K. Galbraith [For admin and other information see: http://www.mta.ca/~cat-dist/ ]