Example 3.5.8.2 (055M): The Canonical Tower

Example 3.5.8.2 (The Canonical Tower). Let $X$ be a Kan complex. For every integer $n \geq 0$, let $u_{n}$ denote the tautological map from $X$ to its fundamental $n$-groupoid $\pi _{\leq n}(X)$. Since $\pi _{\leq n}(X)$ is also an $(n+1)$-groupoid (Remark 3.5.5.3), Proposition 3.5.6.5 guarantees that $u_{n}$ factors uniquely as a composition $X \xrightarrow {u_{n+1} } \pi _{\leq n+1}(X) \xrightarrow {r_ n} \pi _{\leq n}(X)$. We therefore obtain an inverse system of Kan complexes

\[ \cdots \rightarrow \pi _{\leq 3}(X) \xrightarrow {r_2} \pi _{\leq 2}(X) \xrightarrow {r_1} \pi _{\leq 1}(X) \xrightarrow {r_0} \pi _{\leq 0}(X), \]

Since each $u_{n}$ is bijective on $m$-simplices for $m < n$, the induced map $u: X \rightarrow \varprojlim _{n} \pi _{\leq n}(X)$ is an isomorphism of simplicial sets. It follows from Example 3.5.7.25 that $u$ exhibits the inverse system $\{ \pi _{\leq n}(X) \} _{n \geq 0}$ as a Postnikov tower of $X$, which we will refer to as the canonical Postnikov tower of $X$.