From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/339 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: a characterisation of factorisation systems Date: Sat, 15 Mar 1997 09:49:02 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016901 25259 80.91.229.2 (29 Apr 2009 14:55:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:55:01 +0000 (UTC) To: categories Original-X-From: cat-dist Sat Mar 15 09:49:50 1997 Original-Received: by mailserv.mta.ca; id AA16894; Sat, 15 Mar 1997 09:49:02 -0400 Original-Lines: 36 Xref: news.gmane.org gmane.science.mathematics.categories:339 Archived-At: Date: Fri, 14 Mar 1997 15:26:15 GMT From: Paul-Andre Mellies Dear categorists, I have recently proved an (E,M)-factorisation theorem in the framework of axiomatic rewriting systems: every derivation X -> Y factorises (up to Levy permutation equivalence) into a head reduction X -> Z followed by a non-head reduction Z -> Y. One special difficulty in my case is that I do not define the class M of non-head reductions as a category. So, I need a characterisation of factorisation system (E,M) without any assumption of categoricity of E or M. Here is the statement of the theorem I finally proved: ----------------------------------------------------------------------------- Let E and M be two classes of morphisms in a category C. (E,M) is a factorisation system of C if and only if the four following properties hold: 1. every morphism f in C can be factored as f=me with m in M and e in E, 2. if e is a morphism in E and m is a morphism in M then e is orthogonal to m, 3. if i is an iso left composable to e in E, then ie is in E, 4. if i is an iso right composable to m in M, then mi is in M. ----------------------------------------------------------------------------- I do not know if this characterisation already exists in the litterature on factorisation systems. If it does, please send me the reference to integrate in my paper. People interested in the paper can load it there: http://www.dcs.ed.ac.uk/home/paulm/