Corollary 3.5.3.13 (052C)

Corollary 3.5.3.13. Let $X_{\bullet }: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ be a simplicial set and let $n$ be an integer. Then $X_{\bullet }$ is $n$-coskeletal if and only if it satisfies the following condition for each $m \geq 0$:

$(\ast _ n)$

Let $\operatorname{\mathcal{C}}= \operatorname{{\bf \Delta }}_{ \Delta ^{m} }^{\leq n}$ denote the category of simplices of $\Delta ^ m$ having dimension $\leq n$ (see Construction 1.1.3.9). Then the tautological map

\[ \theta _{m}: X_{m} \rightarrow \varprojlim _{ ([k], \sigma ) \in \operatorname{\mathcal{C}}^{\operatorname{op}} } X_{k} \]

is a bijection.

Proof. For each $n \geq 0$, we can identify $\theta _{m}$ with the restriction map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, X_{\bullet } ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( S_{\bullet }, X_{\bullet } ), \]

where $S_{\bullet }$ denotes the colimit $\varinjlim _{ ( [k], \sigma ) \in \operatorname{\mathcal{C}}} \Delta ^ k$. Using Corollary 1.1.4.8, we can identify $S_{\bullet }$ with the $n$-skeleton $\operatorname{sk}_{n}( \Delta ^ m )$, so the desired result follows from Proposition 3.5.3.10 and Remark 3.5.3.11. $\square$