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/ ]