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Removed the following discussion to the nForum:
Zoran Škoda: But there is much older and more general theorem of Hurewitz: if one has a map $p:E\to B$ and a numerable covering of $B$ such that the restrictions $p^{-1}(U)\to U$ for every $U$ in the covering is a Hurewicz fibration then $p$ is also a Hurewicz fibration. But the proof is pretty complicated. For example George Whitehead’s Elements of homotopy theory is omitting it (page 33) and Postnikov is proving it (using the equivalent “soft” homotopy lifting property).
Todd Trimble: Yes, I am aware of it. You can find a proof in Spanier if you’re interested. I’ll have to check whether the Milnor trick (once I remember all of it) generalizes to Hurewicz’s theorem.
Stephan: I wonder if this trick moreover generalizes (in a homotopy theoretic sense) to categories other that $\Top$; for example to the classical model structure on $Cat$?
Removed the first line
This page will probably have to be renamed something like “fiber bundles are fibrations” once I remember how the trick works in detail.
because this page will certainly not be renamed this way.
Instead I prefixed the page by this pointer:
This page meant to recall the proof of the local recognition of Hurewicz fibrations (see there); but it didn’t and doesn’t.
What is the reference for this? A search of the literature does not reveal anything matching “Milnor slide trick”.
Apart from the ideosyncratic terminology, it seems to be the usual proof, as for instance recalled in the reference that we give here
Then what is the point in keeping this page? Should we not replace references to it with references to https://ncatlab.org/nlab/show/Hurewicz+fibration#LocalRecognition? And then delete it?
Since we effectively cannot delete pages (Richard can) we might think of this page as being the eventual home of an $n$Lab writeup of the local recognition theorem for Hurewicz fibrations. Wouldn’t hurt to have that recorded in a nice fashion.
I can delete it if desired, yes :-).
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