$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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


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


The morphism $f$ is a trivial Kan fibration.


The morphism $f$ is a homotopy equivalence.


The morphism $f$ is a weak homotopy equivalence.

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