From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6236 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Sun, 26 Sep 2010 23:54:44 -0300 (ADT) Message-ID: References: Reply-To: selinger@mathstat.dal.ca (Peter Selinger) NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1285635580 22482 80.91.229.12 (28 Sep 2010 00:59:40 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 28 Sep 2010 00:59:40 +0000 (UTC) Cc: categories@mta.ca (categories) To: baez@math.ucr.edu Original-X-From: majordomo@mlist.mta.ca Tue Sep 28 02:59:39 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P0OXe-0001LP-Td for gsmc-categories@m.gmane.org; Tue, 28 Sep 2010 02:59:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51871) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P0OWo-0005jY-Jn; Mon, 27 Sep 2010 21:58:42 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P0OWj-0004x2-OT for categories-list@mlist.mta.ca; Mon, 27 Sep 2010 21:58:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6236 Archived-At: Hi John, thanks for trying to move the discussion away from terminology and back to actual mathematical matters. In order to focus on the math and not on the terminology, let me today use the word "XXXX" instead of "evil". I don't think the notion used in your examples is general enough. For example, fix some groupoid C, and consider the property of an object x: "x is isomorphic to exactly 3 objects of C". To me, this is clearly XXXX, because it is not invariant under equivalences of C. Yet, according to the definition you used in this email, it extends to a functor C -> {F,T}, and therefore is non-XXXX. For a property P of objects x of a category C, "being invariant under isomorphisms of objects in C" is strictly weaker than "being invariant under equivalences of C". Proof: Clearly, any isomorphism of C can be mapped to an identity of some category C' by some equivalence of categories. Therefore, any property that is invariant under equivalences of categories is invariant under isomorphisms of objects. The above example shows that the converse is not true. I think for XXXXness of structures, a similar refinement is needed. To me, the intuitive concept of XXXX for structures is "cannot be transported along equivalences such that the equivalence becomes structure preserving". To say it more explicitly: if category C has the structure, and category C' is equivalent to C (as a category), then C' can be equipped with a structure in such a way that the equivalence (both directions) is structure preserving. This is a fairly subtle concept, not least because it depends on the precise 2-category in question (to fix what "equivalence" and "structure preserving" means). For example, whether the structure of being "strictly monoidal" is XXXX or not depends on what one means by "structure preserving" (e.g., strict monoidal or strong monoidal functors). The natural transformations need to be specified too, so that one can define "structure preserving equivalence". I tried to give a more general and precise 2-categorical definition on the categories list on January 3, 2010, but I am not sure I got it quite right. I think it was Mark Weber who also pointed out, around the same time, that one person's XXXX concept is another person's non-XXXX concept - in a different 2-category. The fact that being XXXX depends on an ambient 2-category means that it is not a moral judgment, and people should not be offended by it. Some perfectly useful things can be XXXX sometimes, and some perfectly useless things can be non-XXXX. For example, even a not-very-natural property like "there are exactly 3 objects isomorphic to x" can be non-XXXX when viewed in the right 2-category. For example, this is the case in the 2-category of categories, functors, and identity natural transformations. Last comment. Thomas Streicher brought up the example of a fibration P: XX -> BB as a concept that was XXXX but very useful. But I don't think this concept is actually XXXX. Certainly if one thinks of the fibration as a *structure* on BB, then this transports very nicely along equivalences. Namely, given any equivalence BB <--> BB', one can find a fibration P' : XX' -> BB' which is equivalent, as a fibration, to P. Right? So I don't think it is correct to identify the concept of XXXX with "having to talk about equality". Rather, it should be defined in some 2-categorical way. See also Mike Shulman's post from January 4, which discussed this distinction in more depth. -- Peter John Baez wrote: > > 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. > ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]