Kerodon

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Construction 4.5.9.1 (Direct Images). Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every integer $n \geq 0$, we let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n}$ denote the collection of pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$ and $f: \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism for which the composition

\[ \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}\xrightarrow {V} \operatorname{\mathcal{D}} \]

coincides with projection onto the second factor. Note that every nondecreasing function $\alpha : [m] \rightarrow [n]$ induces a map

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

where $f'$ denotes the composite map

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

This construction is compatible with composition, and therefore endows $\{ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})_{n} \} _{n \geq 0}$ with the structure of a simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) = \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{\bullet }$ which we will refer to as the direct image of $\operatorname{\mathcal{E}}$ along $U$.

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