From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1727 Path: news.gmane.org!not-for-mail From: Jiri Rosicky Newsgroups: gmane.science.mathematics.categories Subject: Re: category of fraction and set-theoretic problem Date: Thu, 30 Nov 2000 13:01:51 -0500 (EST) Message-ID: References: <200011300954.KAA08299@irmast2.u-strasbg.fr> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018048 32626 80.91.229.2 (29 Apr 2009 15:14:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:08 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Dec 1 15:15:38 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB1IUjv14510 for categories-list; Fri, 1 Dec 2000 14:30:45 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: rosicky@pascal.math.yorku.ca In-Reply-To: <200011300954.KAA08299@irmast2.u-strasbg.fr> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 58 Original-Lines: 36 Xref: news.gmane.org gmane.science.mathematics.categories:1727 Archived-At: If S is the class of weak equivalences in a Quillen model structure then C[S^{-1}] is always locally small. See, e.g., M. Hovey, Model categories, AMS 1999, Jiri Rosicky On Thu, 30 Nov 2000, Philippe Gaucher wrote: > Bonjour, > > > I have a general question about localizations. > > I know that for any category C, if S is a set of morphisms, then > C[S^-1] exists. And moreover if C is small, then C[S^{-1}] is small > as well (as in the Borceux's book Handbook of categorical algebra I) > > If S is not small, and if we suppose that all sets are in some universe > U, then the previous construction gives a solution as a V-small category > for some universe V with U \in V (the objects are the same but the homsets > need not to be U-small). So it does not work if one wants to get U-small > homsets. > > Another way is to have a calculus of fractions (left or right) and if > S is locally small as defined in Weibel's book "Introduction to homological > algebra". > > But in my case, the Ore condition is not satisfied. Hence the question : > is there other constructions for C[S^{-1}] ? > > > Thanks in advance. pg. > > >