From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1741 Path: news.gmane.org!not-for-mail From: Dan Christensen Newsgroups: gmane.science.mathematics.categories Subject: Re: Can we ignore smallness? Date: 06 Dec 2000 16:49:35 -0500 Message-ID: <873dg15ifk.fsf@jdc.math.uwo.ca> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018059 32687 80.91.229.2 (29 Apr 2009 15:14:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:19 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Dec 7 12:08:24 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB7FPow03123 for categories-list; Thu, 7 Dec 2000 11:25:50 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.7 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 14 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:1741 Archived-At: I agree with what Marco Grandis wrote, suggesting that sometimes it is important to know that the hom sets in a category are small, and want to just supplement what he said with some examples from topology. In topology, one often wants to use a generalized homology or cohomology theory E to compute something, and it can be useful to "localize" a space with respect to this (co)homology theory. The localization X --> L_E X can be characterized as the terminal map from X which induces an isomorphism under E. The existence of such localizations for all X is equivalent to the category Top[(E-isomorphisms)^{-1}] having small hom sets, and so knowing that the latter is true means that one has an important tool for practical computations. The paper by Bousfield Bousfield, A. K. The localization of spaces with respect to homology. Topology 14 (1975), 133--150. is considered quite important because it showed that for any generalized homology theory E, localizations exist, and these localizations now play a central role in homotopy theory. Bousfield proved the existence by showing that the category of fractions above has small hom sets. And he did that by showing that there is a model structure on the category Top with the E-isomorphisms as the weak equivalences. Note that it is still an open question as to whether *co*homological localizations exist for every cohomology theory E! Casacuberta, Scevenels and Jeff Smith have recently shown that they exist if you assume Vopenka's principle, but if anyone can prove this in general or show it is independent of ZFC, that would be considered very interesting. Dan