* A well known result @ 2011-01-14 1:24 JeanBenabou [not found] ` <alpine.LRH.2.00.1101141747190.9206@siskin.dpmms.cam.ac.uk> 0 siblings, 1 reply; 2+ messages in thread From: JeanBenabou @ 2011-01-14 1:24 UTC (permalink / raw) To: Prof. Peter Johnstone, Categories 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 2+ messages in thread
[parent not found: <alpine.LRH.2.00.1101141747190.9206@siskin.dpmms.cam.ac.uk>]
* Re: A well known result [not found] ` <alpine.LRH.2.00.1101141747190.9206@siskin.dpmms.cam.ac.uk> @ 2011-01-14 19:56 ` JeanBenabou 0 siblings, 0 replies; 2+ messages in thread From: JeanBenabou @ 2011-01-14 19:56 UTC (permalink / raw) To: Prof. Peter Johnstone, Categories 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 2+ messages in thread
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