From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5454 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: evil Date: Wed, 30 Dec 2009 10:50:34 -0800 Message-ID: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1262526880 683 80.91.229.12 (3 Jan 2010 13:54:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 3 Jan 2010 13:54:40 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sun Jan 03 14:54:33 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 1NRQuc-0001sz-9j for gsmc-categories@m.gmane.org; Sun, 03 Jan 2010 14:54:30 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NRQXs-0004ce-EN for categories-list@mta.ca; Sun, 03 Jan 2010 09:31:00 -0400 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5454 Archived-At: [Note from moderator: Several messages to categories apparently hung in a mail system for several days. With apologies to posters, I am about to post four from late Decemeber in what should have been their posting order. Sorry about the delay, Bob] A dagger structure on a category should not really be considered evil at all. If you have a functor F: C^op -> C and ask whether it is a dagger structure, then this is (taken literally) an evil question; the answer is yes iff F^2 = 1 and F is the identity on objects, both evil conditions. More precisely, two isomorphic functors may have different answers. (A non-evil version is to ask whether F is isomorphic to a dagger structure.) However, it's not necessary to define a dagger-category as a category C equipped with a functor F: C^op -> C such that F satisfies these conditions. In lower-level language, we ask instead that C be equipped with an operation that takes each morphism f: x -> y to a morphism f^\dag: y -> x such that id^\dag = id, (f g)^\dag = g^\dag f^\dag, and (f^\dag)^\dag = f. Nothing here refers to equality of objects; it can be formulated in a language that (like FOLDS) does not have this concept. Given a dagger structure on C, defined in this elementary way, we can construct a functor \dag: C \to C^op that satisfies the evil property. (Of course, it also satisfies the non-evil version of that property.) But that is neither here nor there as to whether dagger structures are evil. There is some new discussion on the nLab: http://ncatlab.org/nlab/show/evil#daggers In particular, Mike Shulman shows how to translate dagger structures along equivalences of categories, proving that they are not evil. My previous post on this subject should probably be ignored. While any concept ~can~ be de-evilled in the way shown there, this does not necessarily give you the concept that you want, and indeed it need not even preserve already non-evil concepts. (And in this case specifically, it does not seem to be correct, as others have already argued here.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]