# Kerodon

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Construction 5.3.4.7 (The Universal Scaffold). Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories, and let $\operatorname{sTr}_{ \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denote the strict transport representation of $U$ (Construction 5.3.1.5). For each $n \geq 0$, we can identify $n$-simplices of the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}(\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$ with pairs $(\sigma , \tau )$, where $\sigma$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (given by a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$) and $\tau$ is an $n$-simplex of the $\infty$-category $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C_0) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0 / } ), \operatorname{\mathcal{E}})$, which we identify with a morphism of simplicial sets $\Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ C_0/ }) \rightarrow \operatorname{\mathcal{E}}$. Let us identify the diagram $C_0 \xrightarrow {\operatorname{id}} C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ with an $n$-simplex $\widetilde{\sigma }$ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0/} )$, and let $\lambda _{u}( \sigma , \tau )$ denote the $n$-simplex of $\operatorname{\mathcal{E}}$ given by the composite map

$\Delta ^ n \xrightarrow { (\operatorname{id}, \widetilde{\sigma }) } \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0/} ) \xrightarrow {\tau } \operatorname{\mathcal{E}}.$

The construction $(\sigma , \tau ) \mapsto \lambda _{u}( \sigma , \tau )$ determines a morphism of simplicial sets

$\lambda _{u}: \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{E}},$

which we will refer to as the universal scaffold of the cocartesian fibration $U$.