From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6232 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Sun, 26 Sep 2010 11:29:07 +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: dough.gmane.org 1285542162 20170 80.91.229.12 (26 Sep 2010 23:02:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 26 Sep 2010 23:02:42 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Mon Sep 27 01:02:40 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P00Ev-0003Sl-GM for gsmc-categories@m.gmane.org; Mon, 27 Sep 2010 01:02:37 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48684) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P00EC-0002Gi-8w; Sun, 26 Sep 2010 20:01:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P00E8-0004ls-Jb for categories-list@mlist.mta.ca; Sun, 26 Sep 2010 20:01:48 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6232 Archived-At: Dear Andre - > Many good things in mathematics are depending on the choice > of a representation which is not invariant under equivalences, > or under isomorphisms. Modern geometry would not exists > without coordinate systems. I agree. I think you're arguing against a position that nobody here has espoused. A coordinate system is a structure, not a property. In my earlier email I said a *property* is evil if it's not invariant under equivalences. But I'd say a *structure* is evil if it's not *covariant* under equivalences. Coordinate systems are covariant under equivalences, so they're not evil. Let me expand on this a bit, first for properties and then for structures. Say we have a groupoid C. A (possibly evil) property of objects in C is a map F: Ob(C) -> {F,T} where Ob(C) is the class of objects of C and {F,T} is the set of truth values. I say the property is non-evil if it extends to a functor F: C -> {F,T} where now we regard {F,T} as a discrete groupoid. For example, suppose C is the groupoid of vector spaces over your favorite field. A typical non-evil property is "being 5-dimensional". A typical evil property is "having the empty set as its origin". (In ZF set theory, we can take any vector space and ask whether its zero element happens to be the empty set.) I think most mathematicians would be happy to see a theorem that begins Theorem: If V is a vector space that is 5-dimensional... but somewhat surprised to see a theorem that begins: Theorem: If V is a vector space with the empty set as its origin... We instinctively feel that any theorem of the second sort could have been phrased better: how could it really matter that the origin is the empty set? Now, structures. Again, let C be a groupoid. A (possibly evil) structure on objects in C is a map F: Ob(C) -> Set The idea is that for any object c in C, F(c) is the set of structures that can be put on that object. I say the structure is non-evil if it extends to a functor F: C -> Set If C is the groupoid of vector spaces, a typical non-evil structure is a basis: here F(c) is the set of bases of the vector space c. A typical evil structure would be "a basis, if the underlying set of c is the real numbers, but two bases otherwise." Again, I think most mathematicians would feel happy to work with the non-evil structure, but somewhat uncomfortable working with the evil one. That's the feeling that this concept of "evil" is trying to formalize. Note that a non-evil structure on finite sets is what you call a "species". You wisely avoided studying the evil ones. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]