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Path: news.gmane.org!not-for-mail
From: =?iso-8859-1?Q?Jean_B=E9nabou?= <jean.benabou@wanadoo.fr>
Newsgroups: gmane.science.mathematics.categories
Subject: A brief survey of cartesian functors
Date: Mon, 28 Jul 2014 11:54:32 +0200
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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/ ]