A remark about Street fibrations

A remark about Street fibrations

[Jean Bénabou]

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In order to stay simple I shall consider only the case where the 2-category with comma objects and 2-pullbacks is Cat, and I shall even accept the axiom of universes and AC.

In that case, but only in that case, the Grothendieck construction makes sense and the theorem, proved by Grothendieck himself, stating that indexed categories and fibered ones are 2-equivalent is, of course, correct.
Nevertheless, as Dubuc points out, Grothendieck discarded indexed categories for fibrations, and neither he nor his school ever used indexed categories.
For ages, I have tried to convince people, in particular on this mailing list, that indexed categories ought to  be abandoned, not only because of the authority of Grothendieck, but for mathematical reasons which I explained, and I wrote the paper in the Journal of Symblic Logic  (JSL) to explain my views.
This had no effect, and sometimes got me a lot of abuse. In particular the JSL paper was qualified as a pamphlet, and I still consider it as the deepest paper I ever wrote. That is why I dedicated it to Grothendieck.
I'm glad to see that in the discussion, only fibrations were considered, and such notions  as fibrations with internal sums and products, which I introduced, were used.

So far, I have had no reaction from Bill Lawvere who introduced indexed categories, and whose influence determined the wrong choice of indexed versus fibered made by many people.
More surprisingly, no reaction either from Peter Johnstone who made in the Elephant a complete mess of the whole question. I hope he will give his opinion.

I would like to thank Ross Street for his answer, and make a few comments on his notion of non evil fibration to answer a question asked by Steve Vickers which I quote:

But that seems to claim that in Cat it doesn't matter whether you use iso or  equality in the Chevalley condition. Does that accord with your understanding?

I disagree totally with the claim. 

Take C = Cat, with all the strong conditions I recalled at the beginning.
I can prove that any functor  p: E -> B  , where  B is a groupoid,  is both a fibration and an op-fibration in the sense of Street.

Let us see some  special cases.
Let B be a groupoid. Take b an object of B,  E an arbitrary category and for  p: E --> B  the constant functor with value b.  you get a Street bifibration with all fibers empty except the fiber over b, which can be chosen arbitrary.
By taking for E a coproduct, you can even have a Street bifibration having over each object of B an arbitrary fiber,  with all these fibers totally unrelated. 
You can even assume the groupoid B to be connected. (In that case, for a Grothendieck fibration all the fibers are isomorphic)
How does this example fit with the geometrical picture evoked by the word fibration? 

In view of this example I suggest that the name of fibrations should be used exclusively for Grothendieck fibrations, the usual ones or their internalizations along the lines I described, and another name, e.g. weak fibrations, be given to the notion defined by Street.
Woud you agree with this, Ross ?

I apologize for sending this mail individually to many people, but it seems that, for reasons unknown, my previous mail has not been forwarded by the category list. I hope this one will be but I prefer to be on the safe side as I hope to get a few reactions.

Thanks to all for your  patience.

Jean




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Re: A remark about Street fibrations

[Ross Street]

On 23 Jul 2014, at 3:42 pm, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:

> In view of this example I suggest that the name of fibrations should be used exclusively for Grothendieck fibrations, the usual ones or their internalizations along the lines I described, and another name, e.g. weak fibrations, be given to the notion defined by Street.
> Woud you agree with this, Ross ?

Dear Jean

Yes, I do agree that ``weak fibration'' or ``pseudo fibration’’
would be good terminology.
The ``pseudo-fibres” rather than the strict fibres are the relevant concept.

There is no doubt that Grothendieck fibrations are very important.
But the pseudo-fibrations do have a mathematical role, as
outlined is a message sent to ``categories’’ (by a group of us)
a year or two ago. 

Best wishes,
Ross

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Re: A remark about Street fibrations

[Jean Bénabou]

Dear Ross,

I agree with you on all points: 
Pseudo-fibrations is much better than weak fibrations, it describes more precisely the notion defined. I hope it will be adopted by the category-commmunity.
I am also convinced that these pseudo fibrations have a mathematical role.

The example I gave has two interests:
(i) To show that calling them fibrations would violate all our intuitions about the idea of fibration. 
(ii) to provide a new, important,  and perhaps not known example of such pseudo fibrations.

Of course what makes things work in that example is that, if G is a groupoid, for cospans with codomain G comma objects and pseudo pullbacks coincide.
This is also true in any 2-Category C, if you define a groupoid of C to be an object G such that for each object X of C, the category  C(X,G) is a groupoid.

Best wishes,

Jean


Le 24 juil. 2014 à 00:03, Ross Street a écrit :

> On 23 Jul 2014, at 3:42 pm, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
> 
>> In view of this example I suggest that the name of fibrations should be used exclusively for Grothendieck fibrations, the usual ones or their internalizations along the lines I described, and another name, e.g. weak fibrations, be given to the notion defined by Street.
>> Woud you agree with this, Ross ?
> 
> Dear Jean
> 
> Yes, I do agree that ``weak fibration'' or ``pseudo fibration’’
> would be good terminology.
> The ``pseudo-fibres” rather than the strict fibres are the relevant concept.
> 
> There is no doubt that Grothendieck fibrations are very important.
> But the pseudo-fibrations do have a mathematical role, as
> outlined is a message sent to ``categories’’ (by a group of us)
> a year or two ago. 
> 
> Best wishes,
> Ross



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