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Proposition Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. The following conditions are equivalent:


The morphism $f$ is a trivial Kan fibration.


The morphism $f$ is a homotopy equivalence.


The morphism $f$ is a weak homotopy equivalence.


For every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a contractible Kan complex.

Proof. The implication $(1) \Rightarrow (2)$ is Proposition, the implication $(2) \Rightarrow (3)$ follows from Proposition, the equivalence $(3) \Leftrightarrow (4)$ is a special case of Corollary, and the equivalence $(4) \Leftrightarrow (1)$ follows from Proposition $\square$