TITLE: A conceptual proof that local fibrations over paracompact spaces are global fibrations?
QUESTION [17 upvotes]: I'm teaching some homotopy theory at the moment, and I'm discovering a number of things I've never before learned properly. (I guess everybody who's ever taught knows that feeling.) One of those things is that a local Hurewicz fibration is a global Hurewicz fibration if the base space is paracompact. This result goes back to Hurewicz and Huebsch, is proved in Spanier's book, and there's a paper by Dold where he compares various local and global properties of projection maps, including being a fibration. None of these proofs are what I would call conceptual. There's a lot of gluing together of "extended" lifting functions, dividing up intervals in three parts, etc etc, and they use the axiom of choice. Since paracompactness needs to get used, any possible proof will probably require some gluing, but does anyone have a more conceptual proof of the result? Has anyone taught this to students, and how was that? Is the axiom of choice a necessary input?

REPLY [4 votes]: You should look at 
Dyer, Eldon and Eilenberg, Samuel. Globalizing fibrations by schedules, Fund. Math., 130, (1988), 125--136. 
They deduce the theorem on globalising fibrations from a more general theorem about the existence of a continuous function $f$ from the path space of $X$ to a space of "schedules" on $X$, given a numerable cover on $X$,  such that any path $a$  "fits" the schedule $f(a)$. The intuitive interpretation is as follows: we know by the Lebesgue covering lemma, that for an open cover of $X$ and a path in $X$ we can find a subivision of the path so that each part of the subdivision lies in a set of the cover.  The aim is to do this trick globally over all paths and "continuously". 
I once tried to follow Spanier's proof in the first edition of his book,but found that a function given in a key proposition was not well defined. I managed to find and send him a definition which was well defined, see the second edition, but I could not prove it was continuous! (He does not give a proof of continuity.)  Probably I'm not clever enough?