Example 3.5.8.5 (The Weakly Coskeletal Tower). Let $X$ be a Kan complex. For every integer $n \geq 0$, let $v^{\circ }_{n}: X \rightarrow \operatorname{cosk}^{\circ }_{n}(X)$ denote the tautological map from $X$ to its weak $n$-coskeleton (Notation 3.5.4.19). Since the $\operatorname{cosk}^{\circ }_{n}(X)$ is $(n+1)$-coskeletal, the morphism $v^{\circ }_ n$ factors (uniquely) as a composition $X \xrightarrow {v^{\circ }_{n+1}} \operatorname{cosk}^{\circ }_{n+1}(X) \xrightarrow {q^{\circ }_ n} \operatorname{cosk}^{\circ }_{n}(X)$. We therefore obtain an inverse system of Kan complexes
which we will refer to as the weakly coskeletal tower of $X$. Since $v^{\circ }_{n}$ is bijective on $m$-simplices for $m < n$, the induced map $v^{\circ }: X \rightarrow \varprojlim _{n} \operatorname{cosk}^{\circ }_{n}(X)$ is an isomorphism of simplicial sets. It follows from Example 3.5.7.24 that $v^{\circ }$ exhibits the weakly coskeletal tower as a Postnikov tower of $X$ (that is, each of the morphisms $v^{\circ }_{n}: X \rightarrow \operatorname{cosk}^{\circ }_{n}(X)$ exhibits $\operatorname{cosk}^{\circ }_{n}(X)$ as an $n$-truncation of $X$).