Example 3.5.8.4 (055P): The Coskeletal Tower

Example 3.5.8.4 (The Coskeletal Tower). Let $X$ be a Kan complex. For every integer $n$, let $v_{n}: X \rightarrow \operatorname{cosk}_{n}(X)$ denote the tautological map from $X$ to its $n$-coskeleton. Since the $\operatorname{cosk}_{n}(X)$ is $(n+1)$-coskeletal, the morphism $v_ n$ factors (uniquely) as a composition $X \xrightarrow {v_{n+1}} \operatorname{cosk}_{n+1}(X) \xrightarrow {q_ n} \operatorname{cosk}_{n}(X)$. We therefore obtain an inverse system of Kan complexes

\[ \cdots \rightarrow \operatorname{cosk}_{4}(X) \xrightarrow {q_3} \operatorname{cosk}_{3}(X) \xrightarrow { q_2} \operatorname{cosk}_{2}(X) \xrightarrow {q_1} \operatorname{cosk}_{1}(X) \]

which we will refer to as the coskeletal tower of $X$. Since $v_{n}$ is bijective on $m$-simplices for $m \leq n$, the induced map $v: X \rightarrow \varprojlim _{n} \operatorname{cosk}_{n}(X)$ is an isomorphism of simplicial sets. It follows from Example 3.5.7.23 that $v$ exhibits the coskeletal tower as a Postnikov tower of $X$ (that is, each of the morphisms $v_{n+1}: X \rightarrow \operatorname{cosk}_{n+1}(X)$ exhibits $\operatorname{cosk}_{n+1}(X)$ as an $n$-truncation of $X$).