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I recall having seen discussion of a Hurewicz or Serre fibration equipped with a chosen path lifting function. Citation??

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Perhaps what you're referring to is Section 7.2 (page 57 of the pdf) of Peter May's A Concise Course in Algebraic Topology. Here he characterizes what it means to be a Hurewicz fibration in terms of path lifting functions. Is that what you were asking for?

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  • $\begingroup$ Yes, that was what I was after though not completely. Peter cits no earlier ressult. $\endgroup$
    – jim stasheff
    Commented Nov 25, 2023 at 23:06
  • $\begingroup$ Moreover, I dimly recall somebody working with pairs (fibrati0on, liftingfunction) Thanks $\endgroup$
    – jim stasheff
    Commented Nov 25, 2023 at 23:08
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(Not an answer, but long for a comment.) Spanier's "Algebraic Topology", Section 2.7, gives Hurewicz' proof of the theorem that a local (Hurewicz) fibration with respect to a numerable open cover of the base is a fibration, using lifting functions. He also mentions the proof by Huebsch, which appeared in the same year, and later work by Dold. It is likely that the paper you have in mind references one of these, so perhaps you can look at MathSciNet under Citations/From References to find the paper you have in mind? (Presently there are 32, 4 and 138 of those, respectively.)

MR0073987 (17,519e) Hurewicz, Witold On the concept of fiber space. Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 956–961.

MR0091470 (19,974d) Huebsch, William On the covering homotopy theorem. Ann. of Math. (2) 61 (1955), 555–563.

MR0155330 (27 #5264) Dold, Albrecht Partitions of unity in the theory of fibrations. Ann. of Math. (2) 78 (1963), 223–255.

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