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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