# Kerodon

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Proposition 3.3.7.4. Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. The following conditions are equivalent:

$(1)$

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

$(2)$

The morphism $f$ is a trivial Kan fibration.

$(3)$

The morphism $f$ is a homotopy equivalence.

$(4)$

The morphism $f$ is a weak homotopy equivalence.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 3.2.6.15 and the equivalence $(1) \Leftrightarrow (4)$ follows from Corollary 3.3.7.3. The implication $(2) \Rightarrow (3)$ follows from Proposition 3.1.6.10, and the implication $(3) \Rightarrow (4)$ follows from from Proposition 3.1.6.13, $\square$