From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1732 Path: news.gmane.org!not-for-mail From: Dan Christensen Newsgroups: gmane.science.mathematics.categories Subject: Re: localization : more precise question Date: 02 Dec 2000 13:05:01 -0500 Message-ID: <87elzq7l82.fsf@julian.uwo.ca> References: <200012012113.WAA10773@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 1241018052 32646 80.91.229.2 (29 Apr 2009 15:14:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:12 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Dec 2 16:50:25 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB2K5Aa12517 for categories-list; Sat, 2 Dec 2000 16:05:10 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: <200012012113.WAA10773@irmast2.u-strasbg.fr> User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.7 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 68 Xref: news.gmane.org gmane.science.mathematics.categories:1732 Archived-At: Philippe Gaucher writes: > 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. > > A morphism f of C is in S if and only if f induces an homeomorphism on > the underlying topological spaces. > > 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. Call the category you describe D. There is an obvious functor C --> D which inverts the morphisms of S. So there is an induced functor C[S^{-1}] --> D, which is the identity on objects and is clearly full, since the morphisms from A to B in C[S^{-1}] can be described as the *formal* composites g_1.f_1^{-1}.....f_n^{-1}.g_{n+1} (modulo certain relations), where g_1,...,g_{n+1}, are morphisms of C and f_1,...,f_n morphisms in S. The question is whether the functor C[S^{-1}] --> D is faithful. I suspect that this is true in general, but can only prove it if you restrict yourself to CW-complexes with a finite number of cells. If you do this, then I believe that the reverse Ore condition holds: given s A ---> B | v C with s in S, there exists s A ---> B | | v v C ---> D t with t in S. (B is just A with a finite number of vertices added; just add the images of those points in C as new vertices to get D.) With this, it isn't hard to see that the functor is faithful. For infinite CW-complexes, this Ore condition doesn't hold, but I still suspect that the functor is faithful. In part it depends upon what you mean by "orientation preserving". Does this mean "having a 'positive' derivative at all times"? Or 'non-negative'? Or can the map go forwards and backwards as long as overall it has degree one? Dan