From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5249 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: intersections of classes Date: Thu, 12 Nov 2009 10:45:30 -0200 Message-ID: References: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: lo.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 1258047446 21944 80.91.229.12 (12 Nov 2009 17:37:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 17:37:26 +0000 (UTC) Cc: categories@mta.ca To: Andrew Salch Original-X-From: categories@mta.ca Thu Nov 12 18:37:19 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N8dbg-0006aD-B6 for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 18:37:16 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8d5M-00015Q-Oi for categories-list@mta.ca; Thu, 12 Nov 2009 13:03:52 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5249 Archived-At: There should not be any problem taking intersections as well as unions of large classes indexed by large classes. Problems arise with products or coproducts in categories, but not if the categories are posets. Andrew Salch wrote: > I have a question for the category theorists which is unfortunately just > an issue about sets and classes that I hope some of you have thought about > before, and can help me with: let C be a class, and consider a family of > subclasses C_i of C, which are indexed by an index class I. Am I allowed > to take the intersection of a family of classes indexed by a class? Is the > result a class? > > What I am really thinking of, here, is the situation that C is the class > of objects in an abelian category X; I have two reflective topologizing > subcategories Y,Z of X; and I would like to know that there exists a > smallest reflective topologizing subcategory of X containing both Y and Z. > The intersection of reflective topologizing subcategories is again > reflective and topologizing, so I would like to be able to take the > intersection of all the reflective topologizing subcategories of X > containing both Y and Z (or, what comes to the same thing since all these > subcategories are full subcategories, the full subcategory generated by > the intersection of the object classes of all the reflective topologizing > subcategories of X containing both Y and Z). However this is an > intersection of classes, indexed by a class, and in general one can't > expect any of these classes to be sets. > > When the abelian category X is the category of modules over a commutative > ring, then the class of reflective topologizing subcategories of X forms a > set, so one can take this intersection without any problems; but I do not > suspect that this will be true for all abelian categories. > > More generally, if there is a book or paper on set theory which covers > some of the basic operations you can and can't do with classes, "for the > working mathematician," I'd really like to hear about it. > > Thanks, > Andrew S. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]