From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5077 Path: news.gmane.org!not-for-mail From: Dimitri Ara Newsgroups: gmane.science.mathematics.categories Subject: making a cone universal in a faithful way Date: Mon, 3 Aug 2009 18:37:47 +0200 Message-ID: Reply-To: Dimitri Ara NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1249339297 27815 80.91.229.12 (3 Aug 2009 22:41:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 3 Aug 2009 22:41:37 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Tue Aug 04 00:41:30 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 1MY6Di-0008Hn-EF for gsmc-categories@m.gmane.org; Tue, 04 Aug 2009 00:41:30 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MY5Ok-0000aG-8I for categories-list@mta.ca; Mon, 03 Aug 2009 18:48:50 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5077 Archived-At: Dear List, Has the following elementary problem been already studied? Let C be a category, I a small category, F : I -> C a functor and alpha : F => c a cocone (c is an object of C). When does there exist a category D and a faithful functor G : C -> D taking alpha to a universal cocone? 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. 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. Question: are they sufficient in the general case? Regards, -- Dimitri [For admin and other information see: http://www.mta.ca/~cat-dist/ ]