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Yes, the homotopy fibre of a Kan fibration (of simplicial sets) coincides with the ordinary fibre. This is not supposed to be obvious: it is a consequence of right-properness, which is in turn a consequence of the existence of a very good fibrant replacement functor. In general, given morphisms $f : X \to Z$ and $g : Y \to Z$ in some model category (not ...


I'm going to assume you aren't allowing the edge $\{1,1\}$ which would be a loop and makes it much easier to see that the two aren't homeomorphic. Intuitively, the sub-space topology of Y sort of "smashes" infinitely many of the line segments together, but the simplicial topology of X allows each line segment $\{(1-t)\{1\}+t\{n\}: t>0\}$ to be separated. ...

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