Kerodon

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Definition 5.2.6.12. Let $n \geq 0$ and let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a functor $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$. For $0 \leq i \leq n$, let $\operatorname{\mathcal{C}}(i)$ denote the fiber $\{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}$, which we regard as a full subcategory of $\operatorname{\mathcal{C}}$. A scaffold of $\pi $ is a sequence of functors

\[ \operatorname{\mathcal{C}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{C}}(1) \xrightarrow { F(2) } \operatorname{\mathcal{C}}(2) \xrightarrow { F(3) } \cdots \xrightarrow {F(n)} \operatorname{\mathcal{C}}(n) \]

together with a morphism of simplicial sets

\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}} \]

with the following properties:

$(1)$

For each integer $0 \leq i \leq n$, the composite map

\[ \operatorname{\mathcal{C}}(i) \simeq \{ i \} \times _{\Delta ^ n} M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

coincides with the inclusion of $\operatorname{\mathcal{C}}(i)$ into $\operatorname{\mathcal{C}}$.

$(2)$

For every pair of integers $0 \leq i \leq j \leq n$ and every object $C \in \operatorname{\mathcal{C}}(i)$, the composition

\[ \Delta ^1 \times \{ C\} \rightarrow \operatorname{N}_{\bullet }( \{ i \leq j \} ) \times \operatorname{\mathcal{C}}(i) \rightarrow M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

is a $\pi $-cocartesian edge of $\operatorname{\mathcal{C}}$.