From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5449 Path: news.gmane.org!not-for-mail From: Dusko Pavlovic Newsgroups: gmane.science.mathematics.categories Subject: Re: evil (fwd) Date: Thu, 31 Dec 2009 15:21:19 +0000 (GMT) Message-ID: References: <20091231024138.321955C285@chase.mathstat.dal.ca> Reply-To: Dusko Pavlovic NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1262306585 16708 80.91.229.12 (1 Jan 2010 00:43:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 1 Jan 2010 00:43:05 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Fri Jan 01 01:42:58 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 1NQVbU-0005Wa-BB for gsmc-categories@m.gmane.org; Fri, 01 Jan 2010 01:42:56 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NQVKW-00014Z-Vx for categories-list@mta.ca; Thu, 31 Dec 2009 20:25:25 -0400 In-Reply-To: <20091231024138.321955C285@chase.mathstat.dal.ca> Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5449 Archived-At: the definition that i proposed was overconstrained. it should be: >> DEF. Let CC be a monoidal category. a *dagger* on CC is a functor >> P: CC^op ---> CC which is >> >> * self-adjoint >> * equivalence >> * given together with the maps (*not* necessarily dinatural) >> ** e_X : X (x) PX ---> I >> ** h_X : I--->PX (x) X >> which make PX -| X. if e_X and h_X happen to be dinatural, then P is just the star functor. in general, all we know about P is that it is a functor whose object part maps each object to some dual. the idea is that in the case of FHilb, we can take P to be the functor which will take a linear operator f:A-->B to the conjugate of the dual operator. to get the adjoint operator f^dag:B-->A, tranfer Pf:B*-->A* along the isomorphisms A~A* and B~B*, induced by the frobenius structures on A and B. sorry about the mistake, and thanks for pointing it out. -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]