Kerodon

Proof. Let $S$ be a Kan complex; we wish to show that the slice $\infty $-category $\operatorname{\mathcal{S}}_{/S}$ is cartesian closed. Let $(\operatorname{Set_{\Delta }})_{/S}$ denote the category of simplicial sets $X$ equipped with a map $f: X \rightarrow S$ and let $\operatorname{KFib}(S)$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{/S}$ spanned by those objects where $f$ is a Kan fibration. We regard $\operatorname{KFib}(S)$ as a (locally Kan) simplicial category, whose homotopy coherent nerve can be identified with $(\operatorname{\mathcal{S}})_{/S}$ (see Example 5.5.2.23). We will show that the simplicial category $\operatorname{KFib}(S)$ satisfies the hypotheses of Remark 7.7.3.9.

Note that the category $(\operatorname{Set_{\Delta }})_{/S}$ is cartesian closed. If $X$ and $Y$ are simplicial sets equipped with morphisms $f: X \rightarrow S$ and $g: Y \rightarrow S$, then there is an exponential $M = Y^{X}$ in the category $(\operatorname{Set_{\Delta }})_{/S}$, whose $n$-simplices are given by pairs $(\sigma , \rho )$, where $\sigma : \Delta ^ n \rightarrow S$ is an $n$-simplex of $S$ and $\rho : \Delta ^ n \times _{S} X \rightarrow \Delta ^ n \times _{S} Y$ is a morphism which is compatible with the projection to $\Delta ^ n$. To complete the proof, it will suffice to show that if $f$ and $g$ are Kan fibrations, then the projection map $M \rightarrow S$ is also a Kan fibration. Let $\iota : A \hookrightarrow B$ be an anodyne morphism of simplicial sets; we wish to show that every lifting problem

admits a solution. Invoking the universal property of $M$, we can rewrite (7.85) as a lifting problem

Since $g$ is a Kan fibration, we are reduced to showing that the morphism $\iota _{X}$ is a weak homotopy equivalence. This follows from Proposition 3.4.0.2, since $\iota $ is a weak homotopy equivalence and $f$ is a Kan fibration. $\square$