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.
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
But I disagree totally with you when you say: 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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