### Re: Present and future

George Janelidze <janelg <at> telkomsa.net>

2014-08-04 14:52:59 GMT

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

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