Dear Ross, Dear all, In a recent mail I asked Ross if pseudo cartesian functors between pseudo fibrations had been studied. There are many generalizations of fibrations. Pseudo fibrations are only one of them. But there are also prefibrations, defined by Grothendieck, but almost never considered, and pre foliations, which I define here, which generalise greatly pre fibrations. For such pre foliatons, I define cartesian functors and show that they have striking properties, most of which are not known, even in the very special case of fibrations. I thought this brief survey might interest you, in case you decide to study seriously the properties of pseudo cartesian functors. Best regards to all, Jean 1) PRE FOLIATIONS AND FOLIATIONS. 1.1. Notations and first definitions. If P: X -> S is a functor, I denote by V(P), abbreviated by V, the set of vertical maps for P . For every object s of S I denote by X_s the fiber of X over s. In order to deal, not only with fibrations but also with pre fibrations, as defined by Grothendieck, and even with more general notions such as pre foliations and foliations that I shall define, I adopt Grothendieck's definition of cartesian maps, namely: A map k: y -> x of X is cartesian iff for every map f: z ->x such Pf = Pk there exits a unique vertical map v: z ->y such that f = kv. I denote by K(P) , abbreviated by K , the set of these maps. I call hyper cartesian the maps which in the english texts are called cartesian and I denote by H(P), abbreviated by H, the set of these maps. They will play very little role in this brief survey. 1.2. DEFINITION. A functor P: X --> S is a pre foliation iff every map f of X can be factored as f = kv with k in K and v in V. If moreover K is stable by composition, I say that P is a foliation. 1.3. Remarks. (a) pre foliations and foliations are first order notions and can be internalized. (b) with universes and AC Grothendieck showed that his construction worked also for lax functors into Cat and gave pre fibrations. But there is no reindexing, even lax, for pre foliations. (c) Even when K is stable by composition, in many examples H will be strictly contained in K. (d) Foliations need not even be Giraud fibrations (sometimes called Conduché fibrations). (e) There are many significant examples of (pre) foliations which are not (pre) fibrations, but I cannot give them in such a brief survey. 2) CARTESIAN FUNCTORS Let P: X --> S, P': X' --> S and F: X --> X' be functors such that P = P'F. For every object s of S ,I denote by F_s : X_s --> X'_s the functor induced by F on the fibers. I have a general definition of F being cartesian, without any assumption on P and P' and without any reference to cartesian maps, but it uses distributors in an essential manner. I shall not need it in this survey and shall give the definition only when P is a pre foliation, but without any assumption on P'. 2.1. DEFINITION. If P is a pre foliation and P'F = P, I say that F is cartesian iff it satisfies the following two conditions: (i) It preserves cartesian maps, i.e. k in K(P) => Fk in K(P'). (ii) For every f': y' --> F(x) in X' , with y' in X' and x in X, there exist f: y -->x in X, and v': y' --> F(y) in V(P') such that f' = F(f)v'. 2.2. Remarks: (a) Condition (i) goes without saying but (ii) may seem surprising. However if P is a pre fibration, without any assumption on P', (i) => (ii). Moreover this implication characterizes pre fibrations among pre foliations. In particular if both P and P' are pre fibrations, our definition coincides with Grothendieck's. (b) In the literature cartesian functors have been considered mostly when both P and P' are fibrations, and, even in that case, not much has been said about their properties. Compare with the following: 2.3. THEOREM. If P is a pre foliation, P' arbitrary, and F is cartesian, then: (1) F is faithful iff every F_s is. (2) F is full iff every F_s is. (3) F is essentially surjective iff every F_s is. (4) F is final iff every F_s is. (5) F is flat iff every F_s is. (6) F has a left adjoint iff every F_s has. If moreover P is a foliation, then (7) F is conservative iff every F_s is. 2.3. Remarks: I would like to insist on the fact that I assume nothing on P' in the theorem. Most of these results are not known even in the classical case where both P and P' are fibrations.(See e.g. the Elephant) I had proved all these results, in that case, already in 1983, more than 30 years ago, and I intended to add them, with many other things, to the Roisin notes in the book I was writing on fibered categories. But by that time the notes had been circulated, and their content was used with very little, if any, reference to me. I'm glad I kept these results to myself for two reasons: (a) As many of the results in the Roisin notes, they would be now in the Elephant, of course uglily re-indexed, and of course without any reference to me. If anyone doubts that, let me recall that my paper on distributors is not mentioned in the pharaonic bibliography of the Elephant, and neither is my joint note with Roubaud on descent, although both are used in the book! I have personally addressed my last 3 mails on fibrations to Peter Johnstone and have had no reaction so far. I hope this one will be more successful. (b) By a careful and repeated analysis of the proofs, over many years, trying to understand what made them tick, I ended up with the notion of (pre) foliation which generalizes greatly (pre) fibrations and has a lot of significant examples. Of course the previous theorem is a very small sample of what can be said about (pre) foliations. I have made this mail public. I hope it will not have the same fate as the Roisin notes, and if some of it is used full credit will be given to me. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]