# Kerodon

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Definition 9.10.1.20. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be left fibrations of simplicial sets. We will say that a morphism of simplicial sets $f: X \rightarrow X'$ is an equivalence of left fibrations over $S$ if it satisfies the following conditions:

• The morphism $f$ satisfies $q = q' \circ f$; that is, $f$ is a morphism in the category $\operatorname{LFib}(S)$.

• The homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{LFib}(S)}$.

We say that the left fibrations $q: X \rightarrow S$ and $q': X' \rightarrow S$ are equivalent if there exists a morphism $f: X \rightarrow X'$ which is an equivalence of left fibrations over $S$.