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Nov 18, 2023 at 2:56 comment added Philippe Gaucher @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.
Nov 17, 2023 at 20:36 comment added Denis T 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.)
Nov 17, 2023 at 8:13 history asked Philippe Gaucher CC BY-SA 4.0