From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5079 Path: news.gmane.org!not-for-mail From: Lutz Schroeder Newsgroups: gmane.science.mathematics.categories Subject: Re: making a cone universal in a faithful way Date: Tue, 04 Aug 2009 09:38:28 +0200 Message-ID: Reply-To: Lutz Schroeder NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1249391657 23362 80.91.229.12 (4 Aug 2009 13:14:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 4 Aug 2009 13:14:17 +0000 (UTC) To: dimitri.ara@gmail.com, categories Original-X-From: categories@mta.ca Tue Aug 04 15:14:10 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MYJqC-0001Q3-Qq for gsmc-categories@m.gmane.org; Tue, 04 Aug 2009 15:14:08 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MYJ7x-0001BC-Mj for categories-list@mta.ca; Tue, 04 Aug 2009 09:28:25 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5079 Archived-At: Dear Dimitri, > For example, if I is the empty category, the question becomes "when can > you make c an initial object in a faithful way?". If I is the final > category, then the cocone alpha amounts to a morphism f : F(*) -> c and the > question becomes "when can you make f an isomorphism in a faithful way?". > > There are two obvious necessary conditions. > 1) Let f,g : c -> d be two morphisms of C. If f alpha_i = g alpha_i for > every i in I, then we should have f = g.L > 2) Let f : a -> F(i) and g : a -> F(j) be two morphisms of C such that > alpha_i f = alpha_j g. Then for every cocone beta : F => d, we should > have beta_i f = beta_j g. > > In the case of the empty category, the first condition means that for > every object d there is at most one arrow c -> d and the second condition > is void. In the case of the final category, the first condition means > that f is an epi and the second that f is a mono. It is not hard to prove > that in both cases, theses conditions are sufficient. While I'm willing to believe the case of initial objects, the statement is wrong already for the case of isomorphisms. Let's call f a potential isomorphism if there exists a faithful functor (equivalently an embedding) that makes f an isomorphism. Then one has the following property of potential isomorphisms (from my 1999 thesis; in German, I'm afraid): Lemma: Let s be a potential isomorphism, and let f,g,h,j,l,p be morphisms such that fs = sg hg = ks fl = sp. Then kl = hp. Proof: In an extended category where s has a two-sided inverse s^{-1}, we have gs^{-1} = s^{-1}f from fs = sg, and hence kl = kss^{-1}l = hgs^{-1}l = hs^{-1}fl = hs^{-1}sp = hp [] The property of the lemma is not implied by s being both epi and mono (i.e. a bimorphism). It is comparatatively easy to prove this using contrived examples, such as the following. Let the category A consist of objects A, B, C, D, and families of morphisms g_i: A -> A p_i: B -> A s_k: A -> C h_i: A -> D l_i: B -> C f_i: C -> C k_i: B -> D q_i,r: B -> D indexed over i>=0, k>=1, where we identify f_0 and g_0 with the respective identities. Composition is by addition of indices, with the single exception h_0p_0 = r. (This satisfies the associative law, since the exceptional case r occurs only in trivial cases of the law -- it cannot be pre- or postcomposed with a nontrivial morphism, and its two factors do not have proper factorisations.) Then s_1 is a bimorphism but violates the property of the above lemma, and hence is not a potential isomorphism: f_1s_1 = s_1g_1, h_0g_1 = k_0s_1, and f_1l_0 = s_1p_0, but k_0l_0 = q_0 \neq r = h_0p_0. Best regards, Lutz -- -------------------------------------- PD Dr. Lutz Schro"der Senior Researcher DFKI Bremen Safe and Secure Cognitive Systems Cartesium, Enrique-Schmidt-Str. 5 D-28359 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki.de www.dfki.de/sks/staff/lschrode -------------------------------------- ------------------------------------------------------------- Deutsches Forschungszentrum fu"r Ku"nstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Gescha"ftsfu"hrung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 ------------------------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]