Kerodon

Example 7.7.3.3. Let $X$ and $Y$ be simplicial sets and let $M = \operatorname{Fun}(X,Y)$ be the simplicial set introduced introduced in Construction 1.5.3.1. Then the evaluation map $\operatorname{ev}: M \times X \rightarrow Y$ exhibits $M$ as an exponential of $Y$ by $X$ in the category of simplicial sets (Proposition 1.5.3.2). In particular, the category of simplicial sets is cartesian closed. More generally, if $\operatorname{\mathcal{C}}\subseteq \operatorname{Set_{\Delta }}$ is any full subcategory which is closed under the formation of finite products and the operation $(X,Y) \mapsto \operatorname{Fun}(X,Y)$, then $\operatorname{\mathcal{C}}$ is also cartesian closed. For example, the category of Kan complexes $\operatorname{Kan}$ is cartesian closed (Corollary 3.1.3.4), and the category $\operatorname{QCat}$ of $\infty $-category is cartesian closed (Theorem 1.5.3.7).