A brief survey of cartesian functors

A brief survey of cartesian functors

[Jean Bénabou]

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.
























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Re: A brief survey of cartesian functors

[George Janelidze]

Dear Jean,

I remember you talking about foliations more than 20 years ago, but when
exactly is this done? Long before?

No matter what was done first, I think it would nice to compare this
carefully with the results of

[C. Cassidy, M. H?bert, and G. M. Kelly, Reflective subcategories,
localizations, and factorization systems, Journal of Australian Mathematical
Society (Series A), 1985, 287-329].

The seemingly big difference is that the above-mentioned paper is about
reflections, but in fact having the right adjoint is a much weaker
restriction than it seems (in this context).

Since we don't sent attachments to Categories mailing list, I shall send you
that paper separately.

With best regards to all,
George


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Re: A brief survey of cartesian functors

[Jean Bénabou]

Dear George, 

As i mentioned in my mail, it took me many years to develop foliated categories AND cartesian functors to their full extent. The very first approach starts circa 1984 when I had proved important results on cartesian functors between fibered categories and started wondering about possible generalizations.
Thank you for sending me the paper of Cassidy, Herbert and Kelly which I do not know. I shall look at it carefully, but I doubt very much that it will have ANYTHING to do with foliated categories, let alone cartesian functors which are the essential content of my mail.
I shall explain why after I have read the paper you are sending me.

Best regards to all,
Jean


Le 28 juil. 2014 à 12:52, George Janelidze a écrit :

> Dear Jean,
> 
> I remember you talking about foliations more than 20 years ago, but when exactly is this done? Long before?
> 
> No matter what was done first, I think it would nice to compare this carefully with the results of
> 
> [C. Cassidy, M. Hébert, and G. M. Kelly, Reflective subcategories, localizations, and factorization systems, Journal of Australian Mathematical Society (Series A), 1985, 287-329].
> 
> The seemingly big difference is that the above-mentioned paper is about reflections, but in fact having the right adjoint is a much weaker restriction than it seems (in this context).
> 
> Since we don't sent attachments to Categories mailing list, I shall send you that paper separately.
> 
> With best regards to all,
> George
> 

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Re: A brief survey of cartesian functors

[Eduardo J. Dubuc]

Dear Jean, this mail is a valuable contribution to the subject.
Concerning your last comments about crediting it, I suggest you put  a
title like "Comments on fibrations and foliations" or any other of the
sort, and you upload it to the arXiv. In this way, it will have a wider
distribution (its interest is not limited to this list) and even it will
be possible to put it in the references of any paper and book if the
author has a fair perception of your contributions.

best regards  e.d.

On 28/07/14 06:54, Jean B?nabou wrote:
> 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
>

...

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RE: A brief survey of cartesian functors

[Joyal, André]

Dear Jean,

I apologise for my ignorance of your work.

I guess that an equivalence of categories P:X-->S is always a foliation, but not
a fibration, unless it is surjective on objects.

-André
  
__________________________________
From: Jean Bénabou [jean.benabou@wanadoo.fr]
Sent: Monday, July 28, 2014 5:54 AM
To: Categories
Subject: categories: A brief survey of cartesian functors

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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: A brief survey of cartesian functors

[Jean Bénabou]

Dear André

Your guess is quite correct. More generally every full and faithful functor is a foliation. 
Let me call a functor P: X --> S locally full and faithful (lff)  iff for each object  x of X the obvious functor  X/x --> S/P(x)  is full and faithful. Such functors are also characterized by :  Every map of X is hypercartesian.  Thus they are foliatiions. Now every full and faithful functor is lff. Hence is a foliation.
I mentioned in my mail that there are many foliations which are not fibrations, this is a typical example. It shows how much more general than (pre) fibrations  (pre) foliations can be.
To give an easy but important application, let me note that, if X is a groupoid avery functor  P: X --> S, where S is arbitrary, is a foliation, because all the maps of X are isos, hence hypercartesian.

This gives me the opportunity to explain condition (ii) for cartesian functors in a special case. Suppose
P: X --> S,  P' :  X' --> S and F; X --> X'  verify  P = P'F,  where X, X'  and S are groups. Then P and P' are foliations and F preserves cartesian maps. However F need not be cartesian. More precisely F satisfies (ii) iff P and P' have same image in S. In that case the theorem says:  F is a mono (resp an epi)  iff its  restriction  Ker(P) --> Ker(P') is a mono (resp an epi). This would be obviously false without (ii) 
To complete the picture let us see that  (i) => (ii)  when P is a fibration. In that case P is surjective, i.e. Im(P) = S  contains Im(P') . But  P = P'F => Im(P) is contained in Im(P'), hence the equality required.

Thank you for having given me the occasion to explicit some examples, an in particular to show that (ii) is meaningful.

Bien amicalement,
Jean

Le 28 juil. 2014 à 17:53, Joyal, André a écrit :

> Dear Jean,
> 
> I apologise for my ignorance of your work.
> 
> I guess that an equivalence of categories P:X-->S is always a foliation, but not
> a fibration, unless it is surjective on objects.
> 
> -André
> 

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Re: A brief survey of cartesian functors

[George Janelidze]

Dear Jean,

Talking about the comparison, I had in mind mainly the following: the
vertical-cartesian factorization for a fibration is closely related to the
reflective factorization system for a semi-left-exact reflection (one might
vaguely say "they are the same up to an isomorphism under the assumptions
used in both of them").

Concerning the older discussion on fibrations versus indexed categories:
Please believe me that I fully agree with every instance of "fibrations are
better" you mention. Nevertheless I also agree with "indexed categories are
better", in a different sense. The reason I am saying this now is that I
would like to mention semi-left-exact reflections of Cassidy--Hebert--Kelly
and their generalizations as a THIRD APPROACH (I used them independently
calling them "admissible" in Galois theory, first exactly in 1984).

Best regards,
George

--------------------------------------------------
From: "Jean B?nabou" <jean.benabou@wanadoo.fr>
Sent: Monday, July 28, 2014 1:58 PM
To: "George Janelidze" <janelg@telkomsa.net>
Cc: "Ross Street" <street@ics.mq.edu.au>; "Steve Vickers"
<s.j.vickers@cs.bham.ac.uk>; "Lack Steve" <steve.lack@mq.edu.au>; "Peter
Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>; "Eduardo Dubuc"
<edubuc@dm.uba.ar>; "Thomas Streicher"
<streicher@mathematik.tu-darmstadt.de>; "Robert Par?"
<pare@mathstat.dal.ca>; "Marta Bunge" <martabunge@hotmail.com>; "William
Lawvere" <wlawvere@hotmail.com>; "Michael Wright" <mpbw1879@yahoo.co.uk>;
"Categories" <categories@mta.ca>
Subject: Re: A brief survey of cartesian functors

> Dear George,
>
> As i mentioned in my mail, it took me many years to develop foliated
> categories AND cartesian functors to their full extent. The very first
> approach starts circa 1984 when I had proved important results on
> cartesian functors between fibered categories and started wondering about
> possible generalizations.
> Thank you for sending me the paper of Cassidy, Herbert and Kelly which I
> do not know. I shall look at it carefully, but I doubt very much that it
> will have ANYTHING to do with foliated categories, let alone cartesian
> functors which are the essential content of my mail.
> I shall explain why after I have read the paper you are sending me.
>
> Best regards to all,
> Jean
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: A brief survey of cartesian functors

[Jean Bénabou]

Dear George,

I appreciate very much your pioneer work on Galois theories and the developments you and others have given to that work.
I also believe in the role of analogies in mathematics, and I think category theory is the ideal place where one can give the DEEP analogies a mathematical content.
However, in this case, the analogy seems to me totally superficial, namely: two classes A and B of maps in a category X, and the possibility to factor every map f of X as ab, with a in A and b in B. 

This won't go very far since you need some axioms on the pair (A,B) to start proving anything except trivialities. And, I tried to explain in my previous mail, the properties of pairs (E,M) and (V,K) are so radically different that a common denominator would be reduced to almost nothing.

Even more important to me, cartesian functors are a very good notion of morphism between pairs (V,K) and (v',K')  which you can prove non trivial results, the theorem in my mail is only an example of such results. As far as I know there is no notion of morphism between pairs (E,M) and (E',M').

Let me point out some features of cartesian functors F: X --X' , viewed abstractly as morphisms (V,K) --> (V',K')  where V = V(P),  K = K(P),  V' = V(P') and K' = K(P').
1) F preserves vertical end cartesian maps. This is harmless, but F also REFLECTS vertical maps.
2) We assume that every map of X can be factored as kv, but we make no such assumption on X'
3) The very nature of the results: For any important properties, F satisfies globally the property iff it satisfies it fiberwise.

If any reasonable notion of morphism of pairs (E,M) was defined someday would reflection of maps in M be considered? Would one accept that (E',M') should not be a factorization system even in a very weak sense?
And if non trivial results could be obtained about such notion would some kind of fibers play a role?

Sorry George, much as I like unifying notions and theories, I cannot see any real, non trivial, relation between factorization systems and (pre folations + cartesian functors)
I insist on the second term of the previous symbolic addition.
There would be a lot more to say about indexed versus fibered, but you already know my opinion about that. Moreover indexed is totally irrelevant here becausethere is no reindexing for pre foliations

Best regards,
Jean



Le 29 juil. 2014 à 09:02, George Janelidze a écrit :

> Dear Jean,
> 
> Talking about the comparison, I had in mind mainly the following: the vertical-cartesian factorization for a fibration is closely related to the reflective factorization system for a semi-left-exact reflection (one might vaguely say "they are the same up to an isomorphism under the assumptions used in both of them").
> 
> Concerning the older discussion on fibrations versus indexed categories: Please believe me that I fully agree with every instance of "fibrations are better" you mention. Nevertheless I also agree with "indexed categories are better", in a different sense. The reason I am saying this now is that I would like to mention semi-left-exact reflections of Cassidy--Hebert--Kelly and their generalizations as a THIRD APPROACH (I used them independently calling them "admissible" in Galois theory, first exactly in 1984).
> 
> Best regards,
> George



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Re: A brief survey of cartesian functors

[George Janelidze]

Dear Jean,

Thank you for your kind words at the beginning of your message, and I
apologize if what I said about "factorization" and "cartesian" was unclear.

I did not mean to say that there is any connection between factorization
systems and (pre foliations + cartesian FUNCTORS). What I was trying to say,
was only that the following two constructions are essentially the same (up
to an isomorphism):

(a) For a fibration C-->X every morphism f in C factors as f = me, where m
is a cartesian ARROW and e is a vertical arrow (with respect to the given
fibration).

(b) For a semi-left-exact reflection C-->X (in the sense of
Cassidy--Hebert--Kelly) every morphism f in C factors as f = me, where m is
in M, e is in E, E is the class of all morphisms inverted by C-->X, and M is
its orthogonal class (M can also be defined as the class of trivial covering
morphisms in the sense of Galois theory).

I know this might sound trivial to you, but I think it is a fundamental
connection, which should be widely known. And I believe that instead of

"indexed categories versus fibrations"

one should sometimes also consider

"indexed categories versus fibrations versus semi-left-exact reflections"
(this is why I mentioned a "third approach").

Let me also add now: according to Cassidy--Hebert--Kelly, the factorization
mentioned in (b), where E is as in (b), and M is merely its orthogonal
class, also exists under certain assumptions much weaker than
semi-left-exactness.

But again, I never thought that what you do with pre foliations and
cartesian functors is a similar kind of factorization and/or that it is
contained in the Cassidy--Hebert--Kelly paper!

And I hope you have never felt from me any disrespect of your opinions
and/or of your beautiful ideas and results.

Best regards,
George

--------------------------------------------------
From: "Jean B?nabou" <jean.benabou@wanadoo.fr>
Sent: Tuesday, July 29, 2014 11:16 AM
To: "George Janelidze" <janelg@telkomsa.net>
Cc: "Ross Street" <street@ics.mq.edu.au>; "Steve Vickers"
<s.j.vickers@cs.bham.ac.uk>; "Lack Steve" <steve.lack@mq.edu.au>; "Peter
Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>; "Eduardo Dubuc"
<edubuc@dm.uba.ar>; "Thomas Streicher"
<streicher@mathematik.tu-darmstadt.de>; "Robert Par?"
<pare@mathstat.dal.ca>; "Marta Bunge" <martabunge@hotmail.com>; "William
Lawvere" <wlawvere@hotmail.com>; "Michael Wright" <mpbw1879@yahoo.co.uk>;
"Categories" <categories@mta.ca>
Subject: Re: A brief survey of cartesian functors

> Dear George,
>
> I appreciate very much your pioneer work on Galois theories and the
> developments you and others have given to that work.
> I also believe in the role of analogies in mathematics, and I think
> category theory is the ideal place where one can give the DEEP analogies a
> mathematical content.
> However, in this case, the analogy seems to me totally superficial,
> namely: two classes A and B of maps in a category X, and the possibility
> to factor every map f of X as ab, with a in A and b in B.
>
> This won't go very far since you need some axioms on the pair (A,B) to
> start proving anything except trivialities. And, I tried to explain in my
> previous mail, the properties of pairs (E,M) and (V,K) are so radically
> different that a common denominator would be reduced to almost nothing.
>
> Even more important to me, cartesian functors are a very good notion of
> morphism between pairs (V,K) and (v',K')  which you can prove non trivial
> results, the theorem in my mail is only an example of such results. As far
> as I know there is no notion of morphism between pairs (E,M) and (E',M').
>
> Let me point out some features of cartesian functors F: X --X' , viewed
> abstractly as morphisms (V,K) --> (V',K')  where V = V(P),  K = K(P),  V'
> = V(P') and K' = K(P').
> 1) F preserves vertical end cartesian maps. This is harmless, but F also
> REFLECTS vertical maps.
> 2) We assume that every map of X can be factored as kv, but we make no
> such assumption on X'
> 3) The very nature of the results: For any important properties, F
> satisfies globally the property iff it satisfies it fiberwise.
>
> If any reasonable notion of morphism of pairs (E,M) was defined someday
> would reflection of maps in M be considered? Would one accept that (E',M')
> should not be a factorization system even in a very weak sense?
> And if non trivial results could be obtained about such notion would some
> kind of fibers play a role?
>
> Sorry George, much as I like unifying notions and theories, I cannot see
> any real, non trivial, relation between factorization systems and (pre
> folations + cartesian functors)
> I insist on the second term of the previous symbolic addition.
> There would be a lot more to say about indexed versus fibered, but you
> already know my opinion about that. Moreover indexed is totally irrelevant
> here becausethere is no reindexing for pre foliations
>
> Best regards,
> Jean
>

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Re: A brief survey of cartesian functors

[Jean Bénabou]

Dear George,

Thank you for your mail. I see that all my mathematical arguments have not convinced you, and that trying to add more would be useless.
I respect your opinion although I totally disagree with it.

Best regards,
Jean


Le 29 juil. 2014 à 21:58, George Janelidze a écrit :

> Dear Jean,
> 
> Thank you for your kind words at the beginning of your message, and I apologize if what I said about "factorization" and "cartesian" was unclear.
> 
> I did not mean to say that there is any connection between factorization systems and (pre foliations + cartesian FUNCTORS). What I was trying to say, was only that the following two constructions are essentially the same (up to an isomorphism):
> 
> (a) For a fibration C-->X every morphism f in C factors as f = me, where m is a cartesian ARROW and e is a vertical arrow (with respect to the given fibration).
> 
> (b) For a semi-left-exact reflection C-->X (in the sense of Cassidy--Hebert--Kelly) every morphism f in C factors as f = me, where m is in M, e is in E, E is the class of all morphisms inverted by C-->X, and M is its orthogonal class (M can also be defined as the class of trivial covering morphisms in the sense of Galois theory).
> 
> I know this might sound trivial to you, but I think it is a fundamental connection, which should be widely known. And I believe that instead of
> 
> "indexed categories versus fibrations"
> 
> one should sometimes also consider
> 
> "indexed categories versus fibrations versus semi-left-exact reflections" (this is why I mentioned a "third approach").
> 
> Let me also add now: according to Cassidy--Hebert--Kelly, the factorization mentioned in (b), where E is as in (b), and M is merely its orthogonal class, also exists under certain assumptions much weaker than semi-left-exactness.
> 
> But again, I never thought that what you do with pre foliations and cartesian functors is a similar kind of factorization and/or that it is contained in the Cassidy--Hebert--Kelly paper!
> 
> And I hope you have never felt from me any disrespect of your opinions and/or of your beautiful ideas and results.
> 
> Best regards,
> George
> 

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Re: A brief survey of cartesian functors

[Paul Levy]

On 28 Jul 2014, at 10:54, Jean Bénabou wrote:

> 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.

Please tell us your general definition using distributors.

Do any of the results in your Theorem 2.3 hold in this more general  
setting?

Paul

> 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.

--
Paul Blain Levy
School of Computer Science, University of Birmingham
+44 121 414 4792
http://www.cs.bham.ac.uk/~pbl












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R A brief survey of cartesian functors

[Jean Bénabou]

Dear Paul,

I have received many mails concerning the posting you mentioned and I try to answer all of them with precision. This takes a lot of time. I shall answer yours, in detail, about the general definition of cartesian functors in terms of disdtributors very soon. But the precise answer shall be a bit long, so please be patient. 
All I can say now is that, if P: X --> S is a prefoliation the general definition reduces to the simple one I have given (the proof is not trivial).
Moreover some of the results of the theorem require the assumption that P is a prefoliation. Hardly surprising, if you want to prove stronger results you need stronger assumptions. The ones I make are, in a sense, minimal. You'll have I think noticed that for 7) I need P to be a foliation.

Thanks for your interest. Best regards,

Jean


Le 1 août 2014 à 12:35, Paul Levy a écrit :

> 
> On 28 Jul 2014, at 10:54, Jean Bénabou wrote:
> 
>> 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.
> 
> Please tell us your general definition using distributors.
> 
> Do any of the results in your Theorem 2.3 hold in this more general setting?
> 
> Paul
> 
>> 2.3.  THEOREM. If P is a pre foliation, P' arbitrary, and F is cartesian, then:
>> (1)  F is faithful    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.
> 
> --
> Paul Blain Levy
> School of Computer Science, University of Birmingham
> +44 121 414 4792
> http://www.cs.bham.ac.uk/~pbl

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