From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5266 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: intersections of classes Date: Thu, 12 Nov 2009 21:31:52 -0600 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1258141971 6707 80.91.229.12 (13 Nov 2009 19:52:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Nov 2009 19:52:51 +0000 (UTC) To: Andrew Salch , categories@mta.ca Original-X-From: categories@mta.ca Fri Nov 13 20:52:44 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 1N92CJ-0005ch-QB for gsmc-categories@m.gmane.org; Fri, 13 Nov 2009 20:52:43 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N91dO-00047p-1g for categories-list@mta.ca; Fri, 13 Nov 2009 15:16:38 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5266 Archived-At: Andrew Salch wrote: > 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? In ZF, "proper class" means "first-order formula," the formula phi(x) being regarded as a stand-in for "the class of sets x such that phi(x)". In this case the only meaning that can be given to "a family of classes indexed by a class" is that we have a formula with two variables phi(x,y), so that each y indexes "the class of sets x such that phi(x,y)". In this case, as Andrej has said, the intersection of these classes is represented by the formula "for all y, phi(x,y)", so there is no problem. In set-class theories such as NBG and MK, classes are things in their own right (rather than having to be represented by first-order formulas). However, in neither of these theories can a class be an *element* of another class, so "a family of classes indexed by a class" can't mean "a class whose elements are classes" or "a class function whose values are classes"---instead it has to be similar to the meaning in ZF, such as a single class P whose elements are ordered pairs, where each set y indexes the class { x | (x,y) \in P }. Again in these theories there is no trouble with the intersection of such an indexed family of classes, it is just the class { x | (x,y) \in P for all y }. However, in your example: > 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 I don't think that "the class of reflective topologizing subcategories of X" is a legitimate class in the sense of ZF, NBG, or MK. Each reflective topologizing subcategory is itself a proper class, and (in NBG and MK) classes can't be elements of other classes, so there is no class of all reflective topologizing subcategories of X, and thus no class to "index" the intersection you want to take. I think you *can* still define this intersection in MK, however: it is the class { a \in ob(X) | a \in W for any reflective topologizing subcategory W of X that contains both Y and Z } This is allowable in MK because its comprehension axiom for classes allows formulas that can quantify over classes. The comprehension axiom of NBG only allows formulas that do not quantify over classes, so I don't think this construction can be done in NBG. Note that while NBG is conservative over ZFC (it can't prove any new theorems about sets), MK is not. All of these problems go away, of course, if you assume a Grothendieck universe U and replace your sets by U-small sets and your classes by U-large ones. > 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. I'm not sure if this is quite what you're looking for, but my paper "Set theory for category theory" addresses some of these issues (arXiv:0810.1279). Best, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]