A well known result

A well known result

[JeanBenabou]

Dear Peter,

In one of my mails I mentioned the following result, which I thought
to be original:

Proposition: Let P: X --> S be a fibration. The functor P is final
iff all its fibers are connected

  From your answer to that mail, dated December 29, I quote:

"please don't deceive yourself that this is a new result. It is a (very)
special case of the theorem of Street and Walters ("The comprehensive
factorization of a functor", Bull. Amer. Math. Soc. 79, 1973) that the
pair (final functors, discrete fibrations) forms a factorization
structure
on Cat. It's true that this result is not stated in the Elephant (why on
earth should it be?), but the Street--Walters factorization (for
internal
categories) is treated in section B2.5."

I tried to prove that my proposition was a consequence of the theorem
of Street-Walters which you quoted in you mail, but did not succeed.
Then I consulted their original paper, hoping to find there more
details which would help me to find a proof. Again in vain.

I'm quite sure that you're right, and that my inability to get a
proof is entirely due to my mathematical limitations.

Thus I'd really be very grateful, if you'd give me a proof, or even a
sketch of a proof, that my proposition is an easy consequence of the
theorem of Street and Walters.

Many thanks in advance and best regards,
Jean


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Re: A well known result

[JeanBenabou]

Dear Peter,

I stand corrected. My proposition is indeed an immediate consequence  
of Street and Walters. It is also a consequence of much more general  
results of mine on foliated categories which I didn't  mention, and I  
didn't realize that this special case was easy. This is no excuse. I  
was careless, and have to "pay" for this carelessness. this is why I  
make my answer public although your mail was addressed only to me.
There are many mathematical questions I asked you, which you didn't  
answer. I hope this mail will incite you to answer some of them.

Best regards,
Jean

Le 14 janv. 11 à 18:54, Prof. Peter Johnstone a écrit :

> Dear Jean,
>
> The derivation seems simple enough to me. Street and Walters showed  
> that
> a functor is final iff it is orthogonal to the class of discrete
> fibrations. In particular this applies to fibrations which are final
> functors; but any fibration admits a factorization through the
> discrete fibration whose fibres are the connected components of
> the original fibres. Hence, if a fibration is orthogonal to discrete
> fibrations, its fibres must be connected. The converse is similar.
>
> Best regards,
> Peter
>
> On Fri, 14 Jan 2011, JeanBenabou wrote:
>
>> Dear Peter,
>>
>> In one of my mails I mentioned the following result, which I  
>> thought to be original:
>>
>> Proposition: Let P: X --> S be a fibration. The functor P is final  
>> iff all its fibers are connected
>>
>> From your answer to that mail, dated December 29, I quote:
>>
>> "please don't deceive yourself that this is a new result. It is a  
>> (very)
>> special case of the theorem of Street and Walters ("The comprehensive
>> factorization of a functor", Bull. Amer. Math. Soc. 79, 1973) that  
>> the
>> pair (final functors, discrete fibrations) forms a factorization  
>> structure
>> on Cat. It's true that this result is not stated in the Elephant  
>> (why on
>> earth should it be?), but the Street--Walters factorization (for  
>> internal
>> categories) is treated in section B2.5."
>>
>> I tried to prove that my proposition was a consequence of the  
>> theorem of Street-Walters which you quoted in you mail, but did  
>> not succeed. Then I consulted their original paper, hoping to find  
>> there more details which would help me to find a proof. Again in  
>> vain.
>>
>> I'm quite sure that you're right, and that my inability to get a  
>> proof is entirely due to my mathematical limitations.
>>
>> Thus I'd really be very grateful, if you'd give me a proof, or  
>> even a sketch of a proof, that my proposition is an easy  
>> consequence of the theorem of Street and Walters.
>>
>> Many thanks in advance and best regards,
>> Jean



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