Example 2.4.2.2 (0323): Functor Categories

Example 2.4.2.2 (Functor Categories). Let $\operatorname{\mathcal{C}}$ be a category and let $Y: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. For every simplicial set $K$, we let $Y^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the functor given on objects by the formula $Y^{K}(C) = \operatorname{Fun}( K, Y(C) )$. If $X: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is another functor, we let $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( X, Y)_{\bullet }$ denote the simplicial set given by the functor

\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad [n] \mapsto \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( X, Y^{ \Delta ^ n} ). \]

Together with the evident composition maps

\[ \circ : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( Y, Z)_{\bullet } \times \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( X, Z)_{\bullet }, \]

this construction endows $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ with the structure of a simplicial category.