$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 10.2.5.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $C_{\bullet }$ be an augmented semisimplicial object of $\operatorname{\mathcal{C}}$, and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The augmented semisimplicial object $C_{\bullet }$ is $(n-1)$-coskeletal, in the sense of Variant 10.2.5.14.
- $(2)$
For each $k \geq n$, the face cube $\tau _{k}: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ of Construction 10.2.5.18 is a limit diagram in $\operatorname{\mathcal{C}}$.
Proof.
We proceed as in the proof of Proposition 10.2.3.14. Let us identify each $\tau _{k}$ with a functor $\operatorname{N}_{\bullet }( P([k]) )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$, where $P( [k] )$ denotes the collection of all subsets of $\{ 0 < 1 < \cdots < k \} $. Let $P^{\leq n}([k] )$ denote the subset of $P( [k] )$ consisting of subsets of cardinality $\leq n$. Unwinding the definitions, we see that $C_{\bullet }$ is $(n-1)$-coskeletal if and only if the following condition is satisfied for each $k \geq n$:
- $(1_ k)$
The functor $\tau _{k}$ exhibits $C_{k}$ as a limit of its restriction to $\operatorname{N}_{\bullet }( P^{\leq n}( [k] ) )^{\operatorname{op}}$.
Similarly, $(2)$ asserts that the following condition is satisfied for each $k \geq n$:
- $(2_ k)$
The face cube $\tau _{k}: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ of Construction 10.2.5.18 is a limit diagram in $\operatorname{\mathcal{C}}$.
To complete the proof, it will suffice to show that if condition $(1_{\ell } )$ is satisfied for $n \leq \ell < k$, then conditions $(1_ k)$ and $(2_ k)$ are equivalent. Our hypothesis that condition $(2_{\ell } )$ is satisfied for $\ell < k$ guarantees that the functor $\tau _{k} |_{ \operatorname{N}_{\bullet }( P^{\leq k}( [k] ) )^{\operatorname{op}} }$ is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( P^{\leq n}( [k]) )^{\operatorname{op}} \subseteq \operatorname{N}_{\bullet }( P^{\leq k}( [k]) )^{\operatorname{op}}$. The equivalence of $(1_ k)$ and $(2_ k)$ is therefore a special case of Corollary 7.3.8.2.
$\square$