# Kerodon

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Variant 5.5.3.9. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\sigma$ be an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram

$C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \rightarrow C_ n$

in the category $\operatorname{\mathcal{C}}$. Then the fiber product $\Delta ^ n \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ can be identified with the iterated relative join

$((( \mathscr {F}(C_0) \star _{ \mathscr {F}(C_1)} \mathscr {F}(C_1)) \star _{ \mathscr {F}(C_2)} \mathscr {F}(C_2)) \star \cdots ) \star _{\mathscr {F}(C_ n)} \mathscr {F}(C_ n).$

See Exercise 5.2.4.20.