From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1721 Path: news.gmane.org!not-for-mail From: "Prof. T.Porter" Newsgroups: gmane.science.mathematics.categories Subject: Re: category of fraction and set-theoretic problem Date: Thu, 30 Nov 2000 14:34:59 +0000 Message-ID: <3A266593.657FD639@bangor.ac.uk> References: <200011300954.KAA08299@irmast2.u-strasbg.fr> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018045 32604 80.91.229.2 (29 Apr 2009 15:14:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:05 +0000 (UTC) To: Philippe Gaucher , "categories@mta.ca" Original-X-From: rrosebru@mta.ca Thu Nov 30 14:04:19 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eAUHTjK09986 for categories-list; Thu, 30 Nov 2000 13:29:45 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 4.7 [en] (X11; I; FreeBSD 3.3-RELEASE i386) X-Accept-Language: en, fr Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 52 Original-Lines: 49 Xref: news.gmane.org gmane.science.mathematics.categories:1721 Archived-At: 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. Dear All, Philippe's question may be answered in part by looking at the construction by Baues and Dugundji (Trans Amer Math Soc 140 (1969) 239 - 256). Another point is that in the homotopical applications it is not that the Ore condition is satisfied but that it is satisfied up to homotopy that counts. A discussion of this in at least one case is to be found on pages 90 - 111 of the book by Heiner Kamps and myself. (see my homepage for the detailed coordinates if you want. The set theoretic question was looked at by various people including Markus Pfenniger in an unpublished manuscript in 1989. Tim ************************************************************ Timothy Porter Mathematics Division, School of Informatics, University of Wales Bangor Gwynedd LL57 1UT United Kingdom tel direct: +44 1248 382492 home page: http://www.bangor.ac.uk/~mas013 Mathematics and Knots exhibition: http://www.bangor.ac.uk/ma/CPM/