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Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-complex $P^{cof}$ to $P$.

Does someone know a nice explicit description of a trivial fibration $P^{cof} \to P$ ?

Of course, the answer is not unique and nice means nothing mathematically. But I hope that any answer will help me somehow.

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    $\begingroup$ Are you not satisfied by the most obvious thing, i. e. simplicial complex formed by "flat simplices"? Points of this space are formal convex linear combinations of functions in P, map that sends a formal sum to a sum is continuous, and fiber over each point is homeomorphic to the weak product of countable number of Ps. (This is in some sense induced by a canonical homotopy comonoid structure on the interval in category of bi-pointed spaces.) $\endgroup$
    – Denis T
    Commented Nov 17, 2023 at 20:36
  • $\begingroup$ @DenisT 'Obvious' like 'nice' means nothing. I like your idea and I will take it as an answer since it works also by replacing the nondecreasing surjective maps by the nondecreasing homeomorphisms. $\endgroup$
    – Philippe Gaucher
    Commented Nov 18, 2023 at 2:56

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