From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4806 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology in definitions of limits Date: Wed, 21 Jan 2009 10:01:03 -0800 Message-ID: Reply-To: John Baez NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020184 14916 80.91.229.2 (29 Apr 2009 15:49:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:49:44 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Jan 21 19:59:25 2009 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Jan 2009 19:59:25 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LPmyH-0007iP-NR for categories-list@mta.ca; Wed, 21 Jan 2009 19:58:57 -0400 Original-Sender: categories@mta.ca Precedence: bulk X-Keywords: X-UID: 27 Original-Lines: 43 Xref: news.gmane.org gmane.science.mathematics.categories:4806 Archived-At: Dear Categorists - On Tue, Jan 20, 2009 at 11:34 PM, Vaughan Pratt wrote: > Colin McLarty wrote: > >> I often call them "test objects" in talking with students (by analogy with >> "test particles" in General Relativity). I don't think I have ever done it >> in print. > > > From a game-theoretic standpoint one can be either taking the test or > administering it. [..] What you're calling a "test" object there is for > me merely the variable being universally quantified over in the definition > of "all." When I teach limits I call Colin's "test object" a "competitor" to the true limit, or "pretender to the throne", and describe the universal property as saying "whatever you can do, I can do better". This game-theoretic approach to universal properties becomes more interesting when dealing with n-categorical weak limits: the two players take turns making moves. First the proponent picks a cone, then the challenger picks a cone, then the proponent picks a map between cones, then the challenger picks a map between cones, then the proponent picks a map between maps between cones, etc.. This idea is important in opetopic n-categories, and there's also an omega-categorical version - a nice discussion appears starting at the bottom of page 32 of this paper by Makkai: http://www.math.mcgill.ca/makkai/equivalence/equivinpdf/equivalence.pdf "The Hero has to answer each move of the Challenger [...] If Hero can keep it up forever, he wins; otherwise he loses." Best, jb