Kerodon

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Proposition 3.3.7.6. Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. The following conditions are equivalent:

$(1)$: The morphism $f$ is a trivial Kan fibration.
$(2)$: The morphism $f$ is a homotopy equivalence.
$(3)$: The morphism $f$ is a weak homotopy equivalence.
$(4)$: 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 3.1.6.10, the implication $(2) \Rightarrow (3)$ follows from Proposition 3.1.6.13, the equivalence $(3) \Leftrightarrow (4)$ is a special case of Corollary 3.3.7.5, and the equivalence $(4) \Leftrightarrow (1)$ follows from Proposition 3.5.2.1. $\square$