Proposition 4.4.2.14. 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.