Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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 3.2.6.8 and the equivalence $(1) \Leftrightarrow (4)$ follows from Corollary 3.3.7.3. The implication $(2) \Rightarrow (3)$ is a special case of Corollary 3.3.7.3, and the implication $(3) \Rightarrow (4)$ follows from from Proposition 3.1.5.11, $\square$