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Definition 11.10.1.10. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cocartesian fibrations of simplicial sets. We will say that a morphism of simplicial sets $f: X \rightarrow X'$ is an equivalence of cocartesian fibrations over $S$ if it satisfies the following conditions:

  • The morphism $f$ carries $q$-cocartesian edges of $X$ to $q'$-cocartesian edges of $X'$ and satisfies $q = q' \circ f$; that is, $f$ is a morphism in the category $\operatorname{CCart}(S)$.

  • The homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{CCart}(S)}$. That is, there exists another morphism $g: X' \rightarrow X$ in $\operatorname{CCart}(S)$ for which the compositions $g \circ f$ and $f \circ g$ are homotopic to the identity maps $\operatorname{id}_{X}$ and $\operatorname{id}_{X'}$, respectively (as morphisms in the simplicial category $\operatorname{CCart}(S)$).

We say that the cocartesian 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 cocartesian fibrations over $S$: that is, if $X$ and $X'$ are isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{CCart}(S)}$.