Proposition 3.5.3.10 (0529)

Proposition 3.5.3.10. Let $X$ be a simplicial set. For every integer $n$, the following conditions are equivalent:

$(1)$

The simplicial set $X$ is $n$-coskeletal.

$(2)$

For every simplicial set $S$, the restriction map

\[ \theta _{S}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(S, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X ) \]

is a bijection.

Proof. For every nonnegative integer $m > n$, the $n$-skeleton of $\Delta ^ m$ is contained in $\operatorname{\partial \Delta }^ m$, and therefore coincides with with the $n$-skeleton of $\operatorname{\partial \Delta }^ m$. We therefore have a commutative diagram of restriction maps

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(\Delta ^ m, X ) \ar [dr]_{ \theta _{ \Delta ^ m} } \ar [rr] & & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ m, X) \ar [dl]^{ \theta _{ \operatorname{\partial \Delta }^ m} } \\ & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(\Delta ^ m), X). & } \]

If condition $(2)$ is satisfied, then the vertical maps are bijections, so the horizontal map is a bijection as well. Allowing $m$ to vary, we deduce that $X$ is $n$-coskeletal.

We now prove the converse. For every simplicial set $S$, we can identify $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(S, X)$ with the inverse limit of the tower of restriction maps

\[ \cdots \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n+2}(S), X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n+1}(S), X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X). \]

Consequently, to prove $(2)$, it will suffice to show that the restriction map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{m}(S), X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{m-1}(S), X) \]

is a bijection for $m > n$. Using Proposition 1.1.4.12, we can reduce to the case $S = \Delta ^ m$, in which case the statement reduces to the assertion that $X$ is $n$-coskeletal. $\square$