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/