Remark 3.5.8.3 (055N): Uniqueness

Remark 3.5.8.3 (Uniqueness). Let $X$ be a Kan complex and let $Y = \{ Y(n) \} _{n \geq 0}$ be a Postnikov tower of $X$. Then $Y$ is homotopy equivalent to the canonical Postnikov tower of Example 3.5.8.2. More precisely, let $u: X \rightarrow \varprojlim _{n} Y(n)$ be a morphism of simplicial sets which exhibits $Y$ as a Postnikov tower of $X$, given by a compatible system of morphisms $u_{n}: X \rightarrow Y(n)$. We then have a commutative diagram of towers

\[ \xymatrix@R =50pt@C=45pt{ \cdots \ar [r] & \pi _{\leq 3}(X) \ar [r] \ar [d]^{ \pi _{\leq 3}(u_3) } & \pi _{\leq 2}(X) \ar [d]^{ \pi _{\leq 2}(u_2) } \ar [r] & \pi _{\leq 1}(X) \ar [d]^{ \pi _{\leq 1}(u_1) } \ar [r] & \pi _{\leq 0}(X) \ar [d]^{\pi _{\leq 0}(u_0) } \\ \cdots \ar [r] & \pi _{\leq 3}( Y(3) ) \ar [r] & \pi _{\leq 2}( Y(2) ) \ar [r] & \pi _{\leq 1}( Y(1) ) \ar [r] & \pi _{\leq 0}( Y(0) ) \\ \cdots \ar [r] & Y(3) \ar [u] \ar [r] & Y(2) \ar [u] \ar [r] & Y(1) \ar [u] \ar [r] & Y(0), \ar [u] } \]

where the upper vertical maps are homotopy equivalences by virtue of our assumption that $u_{n}$ is $(n+1)$-connective (see Corollaries 3.5.7.14 and 3.5.6.15), and the lower vertical maps are homotopy equivalences by virtue of our assumption that each $Y(n)$ is $n$-truncated (Variant 3.5.7.16).