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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