1 - Let P: C ---> B be a functor. I recall that V(P) denotes the subcategory of C having as maps all vertical maps. I have been working for many years now on the following kind of questions. Given an arbitrary subcategory V of a category C, (i) - When is it of the form V(P) for some P ? (ii) - If it is, what relation is there between all the P's such that V=V(P) ? 2 - Before I go further into these questions, especially (ii) let me examine a very important special case where complete answers to (i) and (ii) are well known, namely when C is a group, the answers of course are: (i) - When V is a normal subgroup of G (ii) - If V is normal, let P: C ---> C/V be the canonical surjection on the quotient , and P': C ---> B' be a functor such that V=V(P') , (I'm not assuming that B' is a group), then there is a unique functor Q: C/V ---> B' such that P'=Q.P , and moreover this Q is faithful . I apologize for such trivialities but they will permit me to comment on 1-(i)and (ii) and to sharpen them 3 - The "functor" P of 2 (i) is surjective, but FOR GROUPS this is equivalent to ; P is a fibration, which immediately suggests the following questions for C an arbitrary category and V a subcategory (i) - If V=V(P) for SOME P, is there a FIBRATION P' such that V=V(P'). If it is not always the case, which V's are of the form V(P) for a FIBRATION P ? And of course 1 (ii) can be modified by asking; what relation is there between all the FIBRATIONS P which have the same category V of vertical maps ? This is now purely a question on fibered categories. (ii) There are many variations on (i) e.g. replacing fibration by prefibration, but even for those who don't like prefibrations, here is another kind of variation: A group is a category with pull-backs, and group homomorphisms preserve pull-backs, so in general one might ask : If C is a category with pull-backs, which V's are of the form V(P) for a functor P WHICH PRESERVES PULL-BACKS ? 4 - Let us return to groups. If C is a group and V a subgroup, not necessarily normal, one can denote by C/V the coset space (say right cosets) it is of course not a group but inherits a rich structure from the action of C on it. Now if C is a category, what does one need to assume on a subcategory V of C to be able to construct an analogous C/V and what structure does it inherit ? I have studied many of he previous questions, and in detail the question 4 for which I defined prefoliated and foliated categories which are, roughly speaking, to categorical prefibrations and fibrations, what topological foliafions are are to fibered spaces. The quotient C/V is then a graph, not a category, with extra structure induced by the action of C, which for obvious reasons I call the "transverse graph" . Maybe some persons might consider this as "futile" mathematics. I shall not try to convince them of the contrary. I'm personally very happy to do this kind of mathematics, and my motto in this matter, and many others, is "live and let live" Greetings to all