Kerodon

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

Lemma 5.1.2.7. Let $q: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $q$ is a trivial Kan fibration.

$(2)$

The morphism $q$ is a left fibration and, for every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a contractible Kan complex.

$(3)$

The morphism $q$ is a right fibration and, 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 immediate, and the converse follows from Lemma 5.1.2.6. The equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument. $\square$