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Definition Let $S$ be a simplicial set. We define (non-full) simplicial subcategories

\[ \operatorname{CCart}(S) \subset (\operatorname{Set_{\Delta }})_{/S} \supset \operatorname{Cart}(S) \]

as follows:

  • Let $X$ be an object of the slice category $(\operatorname{Set_{\Delta }})_{/S}$: that is, a simplicial set equipped with a morphism $q: X \rightarrow S$. Then $X$ belongs to the subcategory $\operatorname{Cart}(S) \subset (\operatorname{Set_{\Delta }})_{/S}$ if and only if $q$ is a cartesian fibration, and $X$ belongs to $\operatorname{CCart}(S) \subset (\operatorname{Set_{\Delta }})_{/S}$ if and only if $q$ is a cocartesian fibration.

  • Let $X$ and $X'$ be objects of $\operatorname{Cart}(S)$. Then $\operatorname{Hom}_{\operatorname{Cart}(S)}(X,X')_{\bullet } = \operatorname{Fun}_{/S}^{\operatorname{Cart}}(X,X')^{\simeq }$ is the core of the $\infty $-category $\operatorname{Fun}_{/S}^{\operatorname{Cart}}(X,X')$ of Notation Similarly, if $X$ and $X'$ are objects of $\operatorname{CCart}(S)$, then $\operatorname{Hom}_{\operatorname{CCart}(S)}(X,X')_{\bullet } = \operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X')^{\simeq }$ is the core of the $\infty $-category $\operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X')$

We let $\mathrm{h} \mathit{\operatorname{Cart}(S)}$ and $\mathrm{h} \mathit{\operatorname{CCart}(S)}$ denote the homotopy categories of the simplicial categories $\operatorname{Cart}(S)$ and $\operatorname{CCart}(S)$, respectively (Construction