Present and future

Present and future

[Jean Bénabou]

Dear Thomas,

As I told you in my previous private mail you are entitled to have your own view, and to make it public. You don't have to submit me anything. I shall of course respect your opinion, even if I disagree with it. (by the way I told exactly the same thing to George Janelidze but, not only I could not convince him, but I had the impression we were living on different planets!). 
Of course, if I do disagree, I shall tell you why I do, and try to convince you by purely mathematical arguments, not by the fact that I consider myself as some kind of owner of fibered categories, in spite of the important developments of this theory which I introduced.
And I promise to study carefully your own arguments,c and to change my views about some questions if you convince me, mathematically.

This is by no means an an answer to your mail. I am preparing a more ambitious mail, where I shall expose my views, not only about fibrations but on other important issues, some of which have not, or very little, been touched by the numerous mails about fibrations exchanged during the last weeks. 
Because of the comprehensive scope of this future mail, I beg you to be patient, i shall need some time.
This future mail shall, in a sense, be addressed to me. I'm getting old, and I need to think a little about what I have done, and what I should have done. (Not only in mathematics of course, but the other domains are between me and me).

Best to all,
Jean

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Re: Present and future

[George Janelidze]

Dear Jean,

I don't want us to live on different planets - so, I am making one more
attempt:

My feeling is that you interpret everything I say as "some kinds of
mathematical objects are better than fibrations" ("some kinds" could be
indexed categories, or pseudo-fibrations, or, say, semi-left exact
reflections). And then you give convincing examples where the language of
fibrations works better, and then you say that you could not convince me.
But:

I NEVER said that any of those concepts is better! All I was trying to say
(more than once) is that all of them, including fibrations, are very
important. Moreover, the relationship between them - which is not exactly an
equivalence - is a very serious mathematical result/discovery/idea, which,
as well as as some other ideas of category theory, helps us to see better
the whole planet of mathematics (on which all recipients of this message
live, I suppose).

By the way, a very 'small part' of the relationship between fibrations and
indexed categories, namely the equivalence between discrete fibrations over
a category C (with small fibres) and functors C^op-->Sets, is already a
fundamental result, is not it? Well, working with discrete fibrations
eliminates sets to a larger extend: e.g. we don't need to think of small
fibres, and we can internalize them (I mean, define discrete fibrations over
an internal category). But does it mean that we should forget about
Set-valued functors?

I know everything I said is trivial for you, but, forgive me, you forced me.

Best regards to all,
George

--------------------------------------------------
From: "Jean B?nabou" <jean.benabou@wanadoo.fr>
Sent: Saturday, August 02, 2014 6:00 PM
To: "Thomas Streicher" <streicher@mathematik.tu-darmstadt.de>; "Eduardo
Dubuc" <edubuc@dm.uba.ar>; "Categories" <categories@mta.ca>
Subject: categories: Present and future

> Dear Thomas,
>
> As I told you in my previous private mail you are entitled to have your
> own view, and to make it public. You don't have to submit me anything. I
> shall of course respect your opinion, even if I disagree with it. (by the
> way I told exactly the same thing to George Janelidze but, not only I
> could not convince him, but I had the impression we were living on
> different planets!).
> Of course, if I do disagree, I shall tell you why I do, and try to
> convince you by purely mathematical arguments, not by the fact that I
> consider myself as some kind of owner of fibered categories, in spite of
> the important developments of this theory which I introduced.
> And I promise to study carefully your own arguments,c and to change my
> views about some questions if you convince me, mathematically.
>
> This is by no means an an answer to your mail. I am preparing a more
> ambitious mail, where I shall expose my views, not only about fibrations
> but on other important issues, some of which have not, or very little,
> been touched by the numerous mails about fibrations exchanged during the
> last weeks.
> Because of the comprehensive scope of this future mail, I beg you to be
> patient, i shall need some time.
> This future mail shall, in a sense, be addressed to me. I'm getting old,
> and I need to think a little about what I have done, and what I should
> have done. (Not only in mathematics of course, but the other domains are
> between me and me).
>
> Best to all,
> Jean
>



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

Re: Present and future

[Jean Bénabou]

Dear George,

When you say
> I don't want us to live on different planets - so, I am making one more attempt:
I agree with you and could repeat word for word this sentence.

But I disagree totally with you when you say:
> My feeling is that you interpret everything I say as "some kinds of mathematical objects are better than fibrations" ("some kinds" could be indexed categories, or pseudo-fibrations, or, say, semi-left exact reflections).And then you give convincing examples where the language of fibrations works better,
I never said, or even hinted, that fibered categories are better than pseudo fibrations or semi-left exact reflections, but only that they are different and, in particular for semi-left exact reflections that the analogy was totally superficial.
And I gave many many mathematical arguments to show how radically DIFFERENT they were.

> and then you say that you could not convince me.
These arguments didn't convince you,and I just stated that fact.
> 
> I NEVER said that any of those concepts is better!
I never reproached you that!

> All I was trying to say (more than once) is that all of them, including fibrations, are very important.
you don't have to convince me of that, except for indexed categories which I consider as a VERY BAD approach to fibered ones. I have for years said so, WITH MATHEMATICAL ARGUMENTS, which have not convinced you, but seem to convince more and more people.

> Moreover, the relationship between them - which is not exactly an equivalence - is a very serious mathematical result/discovery/idea,
Sorry, I shall seem to you very dumb but I don't see much relation between left-exact reflections and fibered categories. But you can easily convince me if you give many MATHEMATICAL arguments showing the two notions are DEEPLY related.

> By the way, a very 'small part' of the relationship between fibrations and indexed categories, namely the equivalence between discrete fibrations over a category C (with small fibres) and functors C^op-->Sets, is already a fundamental result, is not it? Well, working with discrete fibrations eliminates sets to a larger extend: e.g. we don't need to think of small fibres, and we can internalize them (I mean, define discrete fibrations over an internal category).
You made both the question and the answer. Discrete fibrations can be internalized, and this internalization is very important, e.g. in Topos theory, but Set valued functors cannot!
 
> But does it mean that we should forget about Set-valued functors?
Of course not! But category theory has taught us how to generalize CORRECTLY well known notions.
> 
> I know everything I said is trivial for you, but, forgive me, you forced me.
I NEVER said, nor hinted that ANYTHING you said was trivial, even when I disagreed with SOME of the things you said.

Best regards,
Jean



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

Re: Present and future

[George Janelidze]

Dear Jean,

>> All I was trying to say (more than once) is that all of them, including
>> fibrations, are very important.
> you don't have to convince me of that, except for indexed categories which
> I consider as a VERY BAD approach to fibered ones. I have for years said
> so, WITH MATHEMATICAL ARGUMENTS, which have not convinced you, but seem to
> convince more and more people.

You could not convince me because I agree, and, moreover, I knew that even
before we first met (in Predela, Bulgaria). More precisely, I know, from
your remarks, but also independently, many mathematical examples where using
fibrations is infinitely better than using indexed categories. I only insist
on replacing "always very bad" with "sometimes very bad". To explain why I
say "before we first met", let me mention 'my' example: extending
Inassaridze's work on generalized satellites in early 70s, my first step was
exactly to replace indexed categories with fibrations! By the way, talking
about fibrations, why do we never mention Yoneda's regular spans, as defined
in

[N. Yoneda, On Ext and exact sequences, J. Fac. Sci. Tokyo 18, 1960,
507-576]?

> Sorry, I shall seem to you very dumb but I don't see much relation between
> left-exact reflections and fibered categories. But you can easily convince
> me if you give many MATHEMATICAL arguments showing the two notions are
> DEEPLY related.

Forgive me, I said "semi-left-exact" (in the sense of
Cassidy--Hebert--Kelly, or, equivalently, one of versions of "admissible" in
the sense of Galois theory), not "left exact". The mathematical argument
consists of the following observations:

(a) A functor F : X-->Y is a fibration if and only if, for every object x in
X, the induced functor F^x : X/x-->Y/F(x) has a right inverse right adjoint.

(b) This is also true if we replace "fibration" with "pseudo-fibration" and
"right inverse right adjoint" with "fully faithful right adjoint" (that is,
if we "do everything up to an isomorphism").

(c) As follows from (b), whenever X and Y have finite limits, F : X-->Y is a
pseudo-fibration if and only if the induced functor F^1 : X/1-->Y/F(1)
(where 1 a terminal object in X, and so X/1 is isomorphic to X) is a
semi-left-exact reflection.

(d) In particular, if F preserves terminal object (and X and Y have finite
limits), then F is a pseudo-fibration if and only if it is a semi-left-exact
reflection.

I understand that you can tell me that there are a lot of cases where my
'mild' conditions do not hold (e.g. it is the case for 'almost' all discrete
fibrations), but, on the other hand there are so many examples where they
do - including those sent to you in my joint message with Ross Street and
Steve Lack. (Actually it was Steve who once asked me, why don't I mention
fibrations?)

In order to avoid any confusion, let me immediately point out that I never
suggested to anyone to replace fibrations with semi-left-exact reflections
in general.

And since you asked about "deeply related": is not it deep that in many case
thinking of abstract cartesian liftings can be replaced with thinking of
adjoint functors and pullbacks? Note that the pullbacks enter the story when
we calculate the right adjoint of F^x using the right adjoint of F. In fact
if we think that F has an "easy" right adjoint, then we can think of
cartesian liftigs as 'reduced to pullbacks' - which is an additional nice
reason for using the term "cartesian".

With great respect to you and your ideas and results-
George

--------------------------------------------------
From: "Jean B?nabou" <jean.benabou@wanadoo.fr>
Sent: Monday, August 04, 2014 6:33 AM
To: "George Janelidze" <janelg@telkomsa.net>
Cc: "Thomas Streicher" <streicher@mathematik.tu-darmstadt.de>; "Eduardo
Dubuc" <edubuc@dm.uba.ar>; "Categories" <categories@mta.ca>
Subject: Re: categories: Present and future

> Dear George,
>
> When you say
>> I don't want us to live on different planets - so, I am making one more
>> attempt:
> I agree with you and could repeat word for word this sentence.
>
> But I disagree totally with you when you say:
>> My feeling is that you interpret everything I say as "some kinds of
>> mathematical objects are better than fibrations" ("some kinds" could be
>> indexed categories, or pseudo-fibrations, or, say, semi-left exact
>> reflections).And then you give convincing examples where the language of
>> fibrations works better,
> I never said, or even hinted, that fibered categories are better than
> pseudo fibrations or semi-left exact reflections, but only that they are
> different and, in particular for semi-left exact reflections that the
> analogy was totally superficial.
> And I gave many many mathematical arguments to show how radically
> DIFFERENT they were.
>
>> and then you say that you could not convince me.
> These arguments didn't convince you,and I just stated that fact.
>>
>> I NEVER said that any of those concepts is better!
> I never reproached you that!
>
>> All I was trying to say (more than once) is that all of them, including
>> fibrations, are very important.
> you don't have to convince me of that, except for indexed categories which
> I consider as a VERY BAD approach to fibered ones. I have for years said
> so, WITH MATHEMATICAL ARGUMENTS, which have not convinced you, but seem to
> convince more and more people.
>
>> Moreover, the relationship between them - which is not exactly an
>> equivalence - is a very serious mathematical result/discovery/idea,
> Sorry, I shall seem to you very dumb but I don't see much relation between
> left-exact reflections and fibered categories. But you can easily convince
> me if you give many MATHEMATICAL arguments showing the two notions are
> DEEPLY related.
>
>> By the way, a very 'small part' of the relationship between fibrations
>> and indexed categories, namely the equivalence between discrete
>> fibrations over a category C (with small fibres) and functors
>> C^op-->Sets, is already a fundamental result, is not it? Well, working
>> with discrete fibrations eliminates sets to a larger extend: e.g. we
>> don't need to think of small fibres, and we can internalize them (I mean,
>> define discrete fibrations over an internal category).
> You made both the question and the answer. Discrete fibrations can be
> internalized, and this internalization is very important, e.g. in Topos
> theory, but Set valued functors cannot!
>
>> But does it mean that we should forget about Set-valued functors?
> Of course not! But category theory has taught us how to generalize
> CORRECTLY well known notions.
>>
>> I know everything I said is trivial for you, but, forgive me, you forced
>> me.
> I NEVER said, nor hinted that ANYTHING you said was trivial, even when I
> disagreed with SOME of the things you said.
>
> Best regards,
> Jean
>

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