Re: Present and future

["George Janelidze" <janelg@telkomsa.net>]

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
>



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