# Kerodon

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Construction 5.3.4.11 (The Taut Scaffold). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Suppose we are given an $n$-simplex of the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is a pair $(\sigma , \tau )$, where $\sigma$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (given by a diagram $C_0 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$) and $\tau$ is an $n$-simplex of the simplicial set $\mathscr {F}(C_0)$. For $0 \leq i \leq n$, let $\tau _{i}$ denote the composite map

$\Delta ^{i} \hookrightarrow \Delta ^{n} \xrightarrow { \tau } \mathscr {F}(C_0) \rightarrow \mathscr {F}(C_ i).$

We then have a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\tau _0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\tau _1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\tau _2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\tau _ n} \\ \mathscr {F}(C_0) \ar [r] & \mathscr {F}(C_1) \ar [r] & \mathscr {F}(C_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(C_ n). }$

Consequently, we can view the pair $( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ as an $n$-simplex of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. The construction $(\sigma , \tau ) \mapsto ( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ determines a morphism of simplicial sets $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. In the special case where $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is a diagram of $\infty$-categories, we will refer to $\lambda _{t}$ as the taut scaffold of the cocartesian fibration $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.