From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1733 Path: news.gmane.org!not-for-mail From: Philippe Gaucher Newsgroups: gmane.science.mathematics.categories Subject: Re: localization : more precise question Date: Sun, 3 Dec 2000 00:33:28 +0100 (MET) Message-ID: <200012022333.AAA11616@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 1241018053 32648 80.91.229.2 (29 Apr 2009 15:14:13 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:13 +0000 (UTC) To: categories@mta.ca, jdc@julian.uwo.ca Original-X-From: rrosebru@mta.ca Sun Dec 3 10:42:04 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB3E9eI18068 for categories-list; Sun, 3 Dec 2000 10:09:40 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Content-MD5: j4IgY+mMIPsZwfCwzuww1w== 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: 6 Original-Lines: 79 Xref: news.gmane.org gmane.science.mathematics.categories:1733 Archived-At: >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. I believe that you are wrong somewhere. The explanation is in post-scriptum (borrowed from a question in sci.math.research which is not yet posted by now). Or maybe I am wrong in the reasonning ? >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? I meant 'non-negative'. Maybe the definition of the category still needs to be debugged. I don't know. (The motivation of this question was to encode the notion of 1-dimensional HDA up to dihomotopy for those who know the subject in a "true" category such that isomorphism classes represent 1-dimensional HDA up to dihomotopy). "having a 'positive' derivative at all times" would be also sufficient I think. Cheers. pg. PS : The natural conjecture is that C[S^{-1}] is equivalent to the category D 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. If U is a universe containing all sets, let V be a universe with U\in V. The categorical construction of C[S^{-1}] (let us call it "C[S^{-1}]") gives a V-small category. "C[S^{-1}]"(A,B) is generally not a set. To see that, take an object like g_1.f_1^{-1}.g_2 with g_1 and g_2 not invertible in C (this is a reduced form which cannot be simplified in "C[S^{-1}]"). Then replace f_1^{-1} by f_1^{-1} \sqcup Id_{codom(f_1^{-1})} \sqcup Id_{codom(f_1^{-1})} \sqcup... and g_1 by g_1 \sqcup g_1 \sqcup g_1 ... Then "C[S^{-1}]"(dom(g_2),codom(g_1)) has as many elements as the sets of U-small cardinals. Therefore "C[S^{-1}]"(dom(g_1),codom(g_2)) is not a set. The relation between "C[S^{-1}]" and D is as follows. There is a canonical V-small map g : "C[S^{-1}]"(A,B) --> Sets(A,B) and D(A,B) is the quotient of the V-small set "C[S^{-1}]"(A,B) by the V-small equivalence relation "x equivalent to y iff g(x)=g(y)". The above element of "C[S^{-1}]"(dom(g_2),codom(g_1)) are all of them identified by this equivalence relation : it is the reason why the homset from dom(g_2) to codom(g_1) becomes a set. The obvious functor from C-->D does invert the morphisms of S. But one has to prove that for any functor C-->E inverting the morphisms of S, C-->E factorizes through C-->D by a unique functor from D-->E. Such functor C-->E factorizes through "C[S^{-1}]" but for proving the factorization through D, one has to prove that E is a sort of concrete category (a category with a faithful functor to Sets). Of course there is no reason for E to be concrete but because of the functor F:C-->E, Im(F) is not too far from a concrete category. C is a concrete category, constructed with oriented graphs. I never heard about a general way of constructing localizations of concrete categories. Does it exist ?