From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5456 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: Quantum computation and categories Date: Fri, 1 Jan 2010 11:06:15 -0800 Message-ID: References: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1262526933 823 80.91.229.12 (3 Jan 2010 13:55:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 3 Jan 2010 13:55:33 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Sun Jan 03 14:55:26 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NRQvU-00024E-SD for gsmc-categories@m.gmane.org; Sun, 03 Jan 2010 14:55:25 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NRQbf-0004pV-MH for categories-list@mta.ca; Sun, 03 Jan 2010 09:34:55 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5456 Archived-At: Happy New Year! Peter wrote: Consider the following two categories: > > (a) the category of finite dimensional complex vector spaces and linear > maps, and (b) the category of finite dimensional Hilbert spaces and linear > maps. > > Specifically, take Toby's proposal, and consider two different objects A,B > of (b) such that both A and B are two-dimensional Hilbert spaces. Let u:A->B > be some non-unitary isomorphism. Using "u" to stand for a non-unitary morphism! Reminds me of the joke: Teacher: Suppose p is a prime number... Student: But what if it's not? Teacher: Well then it wouldn't be called "p", now, would it! > Then you can easily find an equivalence of categories which identifies both > A and B with the two-dimensional vector space C^2, and which identifies u > with the identity morphism on C^2. At this point, you have not equipped the > category (a) with anything useful, because it does not induce a notion > of unitary map on C^2. > Okay, that's a nice argument. I'm pretty sure Lurie gave me some similar argument: take a dagger-category, try to transport the structure along an equivalence of categories, and get something unacceptable. So it seems that, to define the extra structure of Hilbert spaces (on top of > vector spaces), one needs at least one "evil" concept, be it that of unitary > maps or the dagger structure. > If this is really true (and I think it is), we're pushed towards Mark Weber's idea: dagger-categories are not best thought of as categories but rather something new, based on graphs-with-involution instead of graphs. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]