From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2480 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: quantum logic Date: Wed, 22 Oct 2003 12:01:26 -0400 (EDT) Message-ID: References: <20031020195106.GA2487@math-rs-n03.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018695 4360 80.91.229.2 (29 Apr 2009 15:24:55 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:55 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Thu Oct 23 11:11:00 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Oct 2003 11:11:00 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ACg7r-00041x-00 for categories-list@mta.ca; Thu, 23 Oct 2003 11:07:43 -0300 X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs X-X-Sender: barr@triples.math.mcgill.ca In-Reply-To: <20031020195106.GA2487@math-rs-n03.ucr.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 31 Original-Lines: 65 Xref: news.gmane.org gmane.science.mathematics.categories:2480 Archived-At: I will stick to my perception that if you dealing with Hilbert or Banach spaces isomorphisms should be just that. It makes no difference to the *-autonomous structure anyway. For Banach spaces, if you take as underlying functor the closed unit ball, it has an adjoint. It is not tripleable, however, but C^*-algebras are (with the unit ball underlying functor). On Mon, 20 Oct 2003, Toby Bartels wrote: > Michael Barr wrote in part: > > >After giving the matter some thought, I finally decided that the > >category of Hilbert spaces should have as its morphisms norm-reducing > >linear maps. At the very least that will ensure that an isomorphism is > >an isometry. > > True, but are you begging the question by trying to ensure that? > After all, an invertible bounded linear map is enough to deduce > that Hilbert spaces are isomorphic (even in the sense of isometric), > so why not count those maps as isomorphisms themselves? > > This matter is much bigger than Hilbert spaces, of course; > moving to Banach spaces (a closed category even for arbitrary dimension), > we can even see how, /as/ a closed category, it doesn't really matter! > The question is, what is the forgetful functor from Ban to Set? > Do we take the set of all vectors? or do we take the closed unit ball? > The former corresponds to allowing all bounded linear maps as morphisms, > while the latter corresponds to requiring norm-reducing linear maps. > But in the closed category Ban, the Banach space of morphisms > is, whatever your conventions, the space of all bounded linear maps. > Still, this can be consistent with either choice of hom-SET, > since the closed unit ball in the Banach space of bounded linear maps > is none other than your preferred hom-set of norm-reducing maps. > > Jim Dolan (IIRC) suggested that Ban is more fundamentally a closed category > than a category in the first place. > > We can do this on a more elementary level with metric spaces; > is the hom-set the set of all Lipschitz continuous functions, > or is it only the set of distance-reducing functions? > But unlike with Banach (or Hilbert) spaces, this makes a difference > even to the classification of metric spaces into isomorphism classes. > The question becomes, is an isomorphism of metric spaces > merely a relabelling of points keeping all distances the same, > or does it also allow for a recalibration of ones ruler? > Which is the correct interpretation may depend on the application, > and how absolute -- rather than measured in some unit -- the distances are. > (One can even recalibrate more generously to allow as morphisms > all uniformly continuous maps, or even all continuous maps. > Thus classically one speaks of variously "equivalent" metric spaces, > such as "uniformly equivalent" or "topologically equivalent".) > To get closed categories here, one must restrict to bounded metric spaces; > the analysis is a little more fun than for Banach spaces, > especially with the degeneracy surrounding the initial and terminal spaces. > > > -- Toby > > >