From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2467 Path: news.gmane.org!not-for-mail From: Robert Seely Newsgroups: gmane.science.mathematics.categories Subject: Re: quantum logic Date: Sun, 12 Oct 2003 14:31:11 -0400 (EDT) Message-ID: References: <200310120057.h9C0vK816608@math-cl-n01.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018680 4287 80.91.229.2 (29 Apr 2009 15:24:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:40 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Mon Oct 13 09:43:27 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Oct 2003 09:43:27 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A922k-0006DU-00 for categories-list@mta.ca; Mon, 13 Oct 2003 09:43:22 -0300 In-Reply-To: <200310120057.h9C0vK816608@math-cl-n01.ucr.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 18 Original-Lines: 43 Xref: news.gmane.org gmane.science.mathematics.categories:2467 Archived-At: On Sat, 11 Oct 2003, John Baez wrote: > On a related note: I've repeatedly heard people say something > like "the multiplicative fragment of linear logic is the internal > logic of (closed symmetric?) monoidal categories", but I've never heard > a precise result along these lines. Has anyone worked out a sufficiently > general concept of "the internal logic of a category" or "the > internal logic of a certain 2-category of categories" so that one > could take something like a monoidal category, or a symmetric monoidal > category, or a closed symmetric monoidal category - or maybe the > 2-category of all such - and extract by some systematic method the > corresponding "internal logic"? I'm vaguely imagining some class > of generalizations of the Mitchell-Benabou language of a topos, or > something like that - but I'm really interested in the nonCartesian > case. Hi John - You might want to take a look at the paper by Robin Cockett and me "Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories " in TAC Vol 3 No 5. ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1997/n5/n5.ps As for a general theory - there are plenty of examples, though I don't know if anyone has really made a general theory of this notion of a categorical doctrine, often referred to, and based on a paper of Kock and Reyes from the 70's. But there are many examples (many in the papers Robin and I have written on linearly distributive categories and related structures - visit my webpage if you're interested), which make clear how to go from the internal logic of a category to the category and back. I suggest also you look at our "Introduction to linear bicategories" (MSCS:10(2000)2 pp 165-203), also available on my webpage, for a higher dimensional approach. - all the best, Robert --