Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 1.4.3.1. Let $S_{\bullet }$ and $T_{\bullet }$ be simplicial sets. Then the construction

\[ ([n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times S_{\bullet }, T_{\bullet } ) \]

determines a functor from the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of sets. We regard this functor as a simplicial set which we will denote by $\operatorname{Fun}( S_{\bullet }, T_{\bullet } )$.

Note that, given an $n$-simplex $f$ of $\operatorname{Fun}( S_{\bullet }, T_{\bullet } )$ and an $n$-simplex $\sigma $ of $S_{\bullet }$, we can construct an $n$-simplex $\operatorname{ev}( f, \sigma )$ of $T_{\bullet }$, given by the composition

\[ \Delta ^ n \xrightarrow {\delta } \Delta ^ n \times \Delta ^ n \xrightarrow { \operatorname{id}\times \sigma } \Delta ^ n \times S_{\bullet } \xrightarrow { f } T_{\bullet }. \]

This construction determines a map of simplicial sets $\operatorname{ev}: \operatorname{Fun}( S_{\bullet }, T_{\bullet } ) \times { S_{\bullet } } \rightarrow T_{\bullet }$, which we will refer to as the evaluation map.