From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2474 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: re: quantum logic Date: Mon, 13 Oct 2003 11:10:03 -0400 (EDT) Message-ID: References: <200310122208.h9CM8sf26075@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 1241018685 4322 80.91.229.2 (29 Apr 2009 15:24:45 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:45 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Thu Oct 16 16:48:55 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Oct 2003 16:48:55 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AAE77-0003T9-00 for categories-list@mta.ca; Thu, 16 Oct 2003 16:48:49 -0300 X-X-Sender: barr@triples.math.mcgill.ca In-Reply-To: <200310122208.h9CM8sf26075@math-cl-n01.ucr.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 25 Original-Lines: 24 Xref: news.gmane.org gmane.science.mathematics.categories:2474 Archived-At: I think Rick Blute (+ collaborators) has done some things with this. It is not clear whether you want a self-duality or a *-autonomous category. If you stick to finite dimensional Hilbert spaces, the situation seems simple. If V and W are inner product spaces, then for f, g: V --> W, let f.g = \sum f(v_i).g(v_i) the sum taken over an orthonormal basis. I believe this is invariant to an orthonormal base change and it is obviously positive definite. For infinite dimensional spaces, you would have to stick to f for which \sum f(v_i)^2 < oo. But this isn't a category. It is closed under composition (I think) but certainly lacks identities. This gives rise to something called a nuclear category. The category has all maps and there is sub-non-category of nuclear maps. This all goes back (needless to say) to Grothendieck. If by *-category you just mean self dual, well then Hilbert spaces certainly are that. Self dual categories are a dime a dozen. Just take C x C^op. The amazing thing is that if C is closed, C x C^op is *-autonomous, (assuming C has binary cartesian products). Michael