Kerodon

Construction 3.3.2.5 (The $\operatorname{Ex}$ Functor). Let $X$ be a simplicial set. For every nonnegative integer $n$, we let $\operatorname{Ex}_{n}( X)$ denote the collection of all morphisms of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \rightarrow X$. By virtue of Remark 3.3.2.2, the construction $( [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto (\operatorname{Ex}_{n}(X) \in \operatorname{Set})$ determines a simplicial set which we will denote by $\operatorname{Ex}(X)$. The construction $X \mapsto \operatorname{Ex}(X)$ determines a functor from the category of simplicial sets to itself, which we denote by $\operatorname{Ex}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$.