From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3869 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Teaching Category Theory Date: Mon, 27 Aug 2007 18:04:59 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019571 10644 80.91.229.2 (29 Apr 2009 15:39:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:31 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Aug 28 12:34:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Aug 2007 12:34:58 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IQ2vc-0001kk-6J for categories-list@mta.ca; Tue, 28 Aug 2007 12:24:28 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 20 Original-Lines: 62 Xref: news.gmane.org gmane.science.mathematics.categories:3869 Archived-At: > A rough and ready solution would be to tell them that there is a > distinction between "small" and "large" collections, otherwise known as > "sets" and "proper classes". This would necessitate giving them an > example of a large collection, and I guess the obvious choice is the class > in Russell's Paradox. [...] > > There's probably a better solution involving an axiomatization of the > category of sets (along the lines of the Lawvere-Rosebrugh book), or at > least a listing of some its properties. The question is more when than whether to bring up this distinction. Although the Lawvere-Rosebrugh book doesn't define "large" it does define "small relative to Set" in the antepenultimate paragraph of the book (p.250), namely as "can be parameterized by an object of Set". Near the midpoint of the book (p.130) is the statement of Cantor's theorem X < 2^X with the consequence that Set cannot be parameterized by any of its objects (and hence by the definition on p.250 cannot be small relative to itself). In contrast Borceux in his 3-volume series gets the size issue out of the way on pages 1-4 of Volume 1. CTWM is in between, addressing it on p.22 after covering categories, natural transformations, monics, epis, and zeros. One benefit of getting the distinction out of the way near the beginning is that the students won't feel so mystified when they run across it while reading other category-relevant material (as the better students will). The combinatorics of sets (n^m functions from a set of m elements to a set of n elements, etc., which they definitely should know) in no way prepares one for the possibility of an object larger than any set, for which Cantor's theorem is very helpful. Of the above three positionings, I like CTWM's best: early on, yet not so early as to exaggerate its importance relative to the fundamental concepts of CT. Not to say that CTWM starts out ideally. Spending three pages defining "metacategory" and then defining "category" as "any interpretation of the category axioms within set theory" is impenetrably idiosyncratic for students used to more conventional introductions in their other pure maths courses. Once past the size issue Borceux is much more conventional and direct, except for the relatively mild criticism that his definition of "category" is actually the definition of "locally small category." But that's not at all the stumbling block to understanding that "metacategory" presents, in fact if anything it is helpful not to be distracted at the outset by the prospect of large homobjects. While on the topic of size of homobjects, what drawbacks are there to regarding both the objects and homobjects of any n-category as all lying within the n-th Grothendieck universe for a suitable hierarchy U_1 < U_2 < ... of such (with U_0 = 1)? Although admittedly idiosyncratic, it seems very natural to account for large homobjects in a category C with the explanation that C is really a 2-category, whether or not one is making use of the 2-cells. I can see a methodological objection, namely that there is no logical connection between size and existence of n-cells for a given n. Does it create any actual difficulty somewhere? Vaughan