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Definition 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 $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. We will say that a morphism of simplicial sets $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a scaffold if it satisfies the following conditions:


The diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [rr]^{\lambda } \ar [dr] & & \operatorname{\mathcal{E}}\ar [dl]_{U} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]

is commutative (where the left vertical map is the projection map of Construction


The morphism $\lambda $ carries horizontal edges of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.


For every object $C \in \operatorname{\mathcal{C}}$, the induced map

\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow {\lambda } \{ C \} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]

is a categorical equivalence of simplicial sets.