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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 27th 2021

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 $Cat$?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 4th 2021

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.

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.

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 4th 2021

What is the reference for this? A search of the literature does not reveal anything matching “Milnor slide trick”.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 4th 2021
• (edited Apr 4th 2021)

Apart from the ideosyncratic terminology, it seems to be the usual proof, as for instance recalled in the reference that we give here

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 4th 2021
• (edited Apr 4th 2021)

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?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 4th 2021

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.

1. I can delete it if desired, yes :-).