From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6184 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: are fibrations evil? Date: Fri, 17 Sep 2010 12:36:36 +0800 Message-ID: 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 1284773304 26505 80.91.229.12 (18 Sep 2010 01:28:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 18 Sep 2010 01:28:24 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Sat Sep 18 03:28:23 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 1OwmE0-0000HO-Pn for gsmc-categories@m.gmane.org; Sat, 18 Sep 2010 03:28:20 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49489) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OwmCh-00036f-IQ; Fri, 17 Sep 2010 22:26:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OwmCe-0000yj-FM for categories-list@mlist.mta.ca; Fri, 17 Sep 2010 22:26:56 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6184 Archived-At: Thomas wrote, of fibrations: Sure, they are "evil" but it seems to be beneficial to be "evil" sometimes. > Of course! Anybody who thinks that concepts should be avoided merely because they're "evil" in the technical sense must also think that odd numbers are peculiar and perfect groups are the best groups to study. I'm finding it quite amusing how people are getting worked up over the concept just because its name has moral overtones. An excellent example of an evil but useful concept is the concept of "skeletal category". Every category is equivalent to a skeletal one (assuming choice), but not every category is skeletal. So, this concept is evil. But it's sometimes nice to prove a theorem for all categories by proving it for skeletal categories. We can formalize this a bit. Remember, a property P of objects in an n-category is "evil" if there's an object with that property that is equivalent to an object without that property. It's "non-evil" if whenever x has property P and x is equivalent to y, y also has that property. (I'm using classical logic here. If we're using intuitionistic logic, the really useful concept is the concept that I'm calling "non-evil". We should probably call it "good": evil is just the absence of good. But "non-evil" is less likely to cause confusion.) Every property P can be made non-evil by defining a new property P' that means "equivalent to something with property P". Sometimes it's more convenient to work with objects with property P than objects with property P'. And note: if the property Q is non-evil, the theorem all objects with property P have property Q automatically implies all objects with property P' have property Q So, for example, if Q is a non-evil property of categories, to prove Q holds for all categories it suffices to prove it for skeletal categories. Or - with a bit more work to fill in the details - if Q is a non-evil property of functors, to prove Q holds for all Street fibrations it suffices to prove it for fibrations. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]