From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1730 Path: news.gmane.org!not-for-mail From: Philippe Gaucher Newsgroups: gmane.science.mathematics.categories Subject: localization : more precise question Date: Fri, 1 Dec 2000 22:13:06 +0100 (MET) Message-ID: <200012012113.WAA10773@irmast2.u-strasbg.fr> Reply-To: Philippe Gaucher NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018050 32633 80.91.229.2 (29 Apr 2009 15:14:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:10 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Dec 2 11:04:59 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB2EQfZ26225 for categories-list; Sat, 2 Dec 2000 10:26:41 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Content-MD5: NPPm/u4YBwp+GuR5icyu9Q== X-Mailer: dtmail 1.3.0 CDE Version 1.3 SunOS 5.7 sun4u sparc Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 3 Original-Lines: 62 Xref: news.gmane.org gmane.science.mathematics.categories:1730 Archived-At: Re-bonjour, Thank you for your answers. My question was very general. So here is the example. I am going to define the category C and the collection of morphisms S, with respect to what I would like to localize. The object of C are the oriented graph. Such an object X is a topological space obtained by choosing a discrete set X^0 and by attaching 1-dimensional cells *with orientations*. It is a 1-dimensional CW-complex with oriented arrows. The morphisms of C are the continuous maps f from X to Y satisfying this conditions : 1) f(X^0)\subset Y^0 2) f is orientation-preserving 3) f is non-contracting in the sense that a 1-cell is never contracted to one point. Remark I : in C, an arrow x--> is not isomorphic to a point. Remark II : an arrow a-->b can be mapped on the loop a-->a with one oriented arrow from a to a. A morphism f of C is in S if and only if f induces an homeomorphism on the underlying topological spaces. Now here is an example of f\in S which is not invertible : a--->b mapped on a-->x-->b This morphism has no inverse in C because the image of x must be equal to a or b by 1) and therefore one of the arrows would be contracted by 2), which contredicts 3). I would like to know if C[S^{-1}] exists or no (in the same universe). The irresistible conjecture is of course that C[S^{-1}] is equivalent to the category whose objects are that of C and whose morphisms from A to B are the subset of C^0(A,B) (the set of continuous maps from A to B) containing all composites of the form g_1.f_1^{-1}.g_2...g_n.f_n^{-1}.g_{n+1} where g_1,...,g_{n+1}, are morphisms of C and f_1,...,f_n morphisms in S. The Ore condition is not satisfied by S because of this example. The Ore condition says that for any s:A-->B in S, and any f:X-->B, there exists t:Y-->X in S and g:Y-->A such that s.g=f.t. Now the counterexample : A is a--->b, B is a-->x-->b with s as above ; X is a-->x with the inclusion f from X in B. Then necessarily Y=X and t=Id. And s.g(x)=b and f.t(x)=x. Thanks in advance. pg.