Definition 3.5.8.1. Let $X$ be a Kan complex. Suppose we are given an inverse system of Kan complexes $Y = \{ Y(n) \} _{n \geq 0}$, which we display as
\[ \cdots \rightarrow Y(3) \rightarrow Y(2) \rightarrow Y(1) \rightarrow Y(0). \]
We say that a morphism of simplicial sets $u: X \rightarrow \varprojlim _{n} Y(n)$ exhibits $Y$ as a Postnikov tower of $X$ if, for every integer $n \geq 0$, the induced map $u_{n}: X \rightarrow Y(n)$ exhibits $Y(n)$ as an $n$-truncation of $Y$: that is, $Y(n)$ is $n$-truncated and $u_ n$ is $(n+1)$-connective (see Definition 3.5.7.19). We say that $Y$ is a Postnikov tower of $X$ if there exists a morphism $u: X \rightarrow \varprojlim _{n} Y(n)$ which exhibits $Y$ as a Postnikov tower of $X$.