From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1723 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: category of fraction and set-theoretic problem Date: Thu, 30 Nov 2000 09:20:52 -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 1241018046 32611 80.91.229.2 (29 Apr 2009 15:14:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:06 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Nov 30 14:07:06 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eAUHVsb30089 for categories-list; Thu, 30 Nov 2000 13:31:54 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs X-Sender: barr@triples.math.mcgill.ca In-Reply-To: <200011300954.KAA08299@irmast2.u-strasbg.fr> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 54 Original-Lines: 52 Xref: news.gmane.org gmane.science.mathematics.categories:1723 Archived-At: It is not clear if you are interested in special cases or in general conditions. If the latter, I cannot help, but here is an example of a special case. But first, I might ask why it matters. Gabriel-Zisman ignores the question and I think they are right to. Every category is small in another universe. Consider the category C of chain complexes from some abelian category. By this I mean bounded below with a boundary operator of degree -1. Arrows are chain maps of degree 0. Let S denote the class of homotopy equivalences and T the class of homology isomorphisms. Then S < T and there is neither a calculus of right or left fractions for either. On the other hand S^{-1}C is equivalent to C/~ in which you have identified homotopic arrows. This is locally small because you leave the objects alone and it is a quotient. From S < T, it follows that T^{-1}C = T^{-1}S^{-1}C = T^{-1}(C/~) and the image of T in C/~ does have a calculus of fractions (both left and right; duality implies that they are equivalent). Thus there is a notion of homotopy calculus of fractions in this case. I have tried, without success, to find a general condition of which this would be a special case. Michael 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. > >