From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8243 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: A brief survey of cartesian functors Date: Mon, 28 Jul 2014 11:54:32 +0200 Message-ID: Reply-To: =?iso-8859-1?Q?Jean_B=E9nabou?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1283) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406554009 15354 80.91.229.3 (28 Jul 2014 13:26:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 28 Jul 2014 13:26:49 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Mon Jul 28 15:26:43 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XBkws-0001b6-J4 for gsmc-categories@m.gmane.org; Mon, 28 Jul 2014 15:26:42 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48693) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XBkwP-0006K9-AO; Mon, 28 Jul 2014 10:26:13 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XBkwO-0002wX-49 for categories-list@mlist.mta.ca; Mon, 28 Jul 2014 10:26:12 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8243 Archived-At: 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.=20 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 =3D Pk there exits a unique vertical map v: z ->y such that f =3D = 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 =3D 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.=20 (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 = Conduch=E9 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.=20 2) CARTESIAN FUNCTORS Let P: X --> S, P': X' --> S and F: X --> X' be functors such that = P =3D 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.=20 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 =3D 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) =3D> 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' =3D = F(f)v'. 2.2. Remarks:=20 (a) Condition (i) goes without saying but (ii) may seem surprising.=20 However if P is a pre fibration, without any assumption on P', (i) =3D> = (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:=20 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!=20 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.=20 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/ ]