Dear George, As I expected, the paper deals with factorization systems (E, M). It has NOTHING , but NOTHING to do with pre foliation, foliations, pre fibrations and fibrations, let alone with cartesian functors. here are a few reasons, I could give many many more. Let me call for short a cartesian system a pair (V, K) where V is the set of vertical maps and K the set of cartesian maps (in the sense of Grothendieck, which I recalled in my mail) of a pre foliation P: X --> S . You can, if you prefer, assume that P is foliation, a pre fibration, or even a fibration. The following remarks work in all cases. . 1. In (E, M) all the isos are both in E and M. In (V, K) the isos need not be in V. In most important cases they will NOT be in V. 2. In (E, M) each of the classes E and M determines the other. In (V,K) V determines K, but K does not determine V. The most extreme case is the following. Suppose X is a groupoid. For every P: X -> S, with S arbitrary every map of is cartesian (an even hyper cartesian) i.e K(P) = X, hence P is a foliation. Take P = id : X --> X and Q: X --> 1 the unique functor. They are both fibrations, K(P) = K(Q) = X , but V(P) consists only of the identities of X whereas V(Q) is the whole of X As a side remark factorization systems don't make any sense on a groupoid, there is only one, whereas foliations or fibratons with domain a groupoid make perfect sense and are even very important. 3. In (E, M) both E and M are stable by composition, whereas K need not be if P is a pre foliation or a pre fibration 4. For every functor P, V(P) satisfies 3 out of 2, i.e. for every commutative triangle in X, if 2 of the maps are vertical so is the third. This is almost never the case for factorization systems. 5. Factorizations in (E, M) are functorial, they need not be in (V,K) for pre foliations , or even pre fibrations How much more do you need? Let me add that in the paper you sent me the authors assume that the category A, where (E, M) lives admits finite limits and all intersections (even large ones) of strong subobjects. This assumption is totally irrelevant in my work on fibrations or (pre) foliations For cartesian functors the situation is even more hopeless. They are special morpphisms in Cat/S. As far as I can see, if A and A' are categories equipped with factorization systems (E, M) and (E', M') no one has seriously defined when a functor F: A --> A' is a morphism for such structures, and if, or when, such definition is given, in view of the differences I pointed out, this definition will be completely different from my definition of cartesian functors. Summarizing, the intersection of the paper you sent me and my previous mail is EMPTY. Many thanks nevertheless for focussing my attention on that paper, which I shall study in detail very shortly. Best regards, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]