Construction 4.5.9.1 (046N)

Construction 4.5.9.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets and let $\operatorname{\mathcal{D}}$ be another simplicial set. For every integer $n \geq 0$, we let $\operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})_{n}$ denote the collection of pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{B}}$ and $f: \Delta ^ n \times _{ \operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a morphism of simplicial sets. Note that every nondecreasing function $\alpha : [m] \rightarrow [n]$ induces a map

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})_{n} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})_{m} \quad \quad (\sigma , f) \mapsto (\alpha ^{\ast }(\sigma ), f'), \]

where $f'$ denotes the composite map

\[ \Delta ^{m} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow { \alpha \times \operatorname{id}} \Delta ^{n} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow {f} \operatorname{\mathcal{D}}. \]

This construction is compatible with composition, and therefore endows $\{ \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}), \operatorname{\mathcal{D}})_{n} \} _{n \geq 0}$ with the structure of a simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$.

Note that the assignment $(\sigma ,f) \mapsto \sigma $ determines a morphism of simplicial sets $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$. Moreover, there is an evaluation map $\operatorname{ev}: \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$, which carries an $n$-simplex $( \widetilde{\sigma }, (\sigma ,f) )$ of the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ to the $n$-simplex of $\operatorname{\mathcal{D}}$ given by the composite map $\Delta ^{n} \xrightarrow { \operatorname{id}\times \widetilde{\sigma } } \Delta ^{n} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow {f} \operatorname{\mathcal{D}}$.