A well known result

[JeanBenabou <jean.benabou@wanadoo.fr>]

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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