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3.5.8 The Postnikov Tower of a Kan Complex

If $X$ is a Kan complex, then its truncations can be arranged into a diagram.

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 $X$.

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 $X$.

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).

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 $X$).

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

\[ \cdots \rightarrow \operatorname{cosk}^{\circ }_{3}(X) \xrightarrow {q^{\circ }_2} \operatorname{cosk}^{\circ }_{2}(X) \xrightarrow { q^{\circ }_1} \operatorname{cosk}^{\circ }_{1}(X) \xrightarrow {q^{\circ }_0} \operatorname{cosk}^{\circ }_{0}(X) \]

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 $X$).

Remark 3.5.8.6. Let $X$ be a Kan complex. Then the Postnikov towers described in Examples 3.5.8.2, 3.5.8.4, and 3.5.8.5 are related by a commutative diagram

3.77
\begin{equation} \begin{gathered}\label{equation:comparison-of-towers} \xymatrix@R =50pt@C=50pt{ \cdots \ar [r] & \operatorname{cosk}_{4}(X) \ar [r] \ar [d] & \operatorname{cosk}_{3}(X) \ar [r] \ar [d] & \operatorname{cosk}_{2}(X) \ar [r] \ar [d] & \operatorname{cosk}_{1}(X) \ar [d] \\ \cdots \ar [r] & \operatorname{cosk}_{3}^{\circ }(X) \ar [r] \ar [d] & \operatorname{cosk}_{2}^{\circ }(X) \ar [r] \ar [d] & \operatorname{cosk}_{1}^{\circ }(X) \ar [r] \ar [d] & \operatorname{cosk}_{0}^{\circ }(X) \ar [d] \\ \cdots \ar [r] & \pi _{\leq 3}(X) \ar [r] & \pi _{\leq 2}(X) \ar [r] & \pi _{\leq 1}(X) \ar [r] & \pi _{\leq 0}(X), } \end{gathered} \end{equation}

where the vertical maps are trivial Kan fibrations (see Proposition 3.5.4.22 and Corollary 3.5.6.14).

Proposition 3.5.8.7. Let $X$ be a Kan complex, let $n \geq 0$ be an integer, and let $v_{n-1}^{\circ }$ denote the tautological map from $X$ to its weak $(n-1)$-coskeleton $\operatorname{cosk}^{\circ }_{n-1}(X)$. Then:

$(1)$

The morphism $v_{n-1}^{\circ }$ factors uniquely as a composition $X \rightarrow \pi _{\leq n}(X) \xrightarrow { f_ n } \operatorname{cosk}_{n-1}^{\circ }(X)$.

$(2)$

The morphism $f_{n}$ exhibits $\operatorname{cosk}_{n-1}^{\circ }(X)$ as a weak $(n-1)$-coskeleton of the fundamental $n$-groupoid $\pi _{\leq n}(X)$.

$(3)$

The morphism $f_{n}$ is a Kan fibration.

$(4)$

Let $x \in X$ be a vertex and set $G = \pi _{n}(X,x)$. Then the fiber $\{ x\} \times _{ \operatorname{cosk}_{n-1}^{\circ }(X) } \pi _{\leq n}(X)$ is isomorphic to the Eilenberg-MacLane space $\mathrm{K}(G,n)$ of Construction 2.5.6.9.

Proof. Since the weak coskeleton $\operatorname{cosk}_{n-1}^{\circ }(X)$ is an $n$-groupoid (Proposition 3.5.5.10), assertion $(1)$ is a special case of Proposition 3.5.6.5. Note that, since $v_{n-1}^{\circ }$ is surjective on $n$-simplices, the morphism $f_ n$ has the same property. Consequently, to prove $(2)$, it will suffice to show that $f_{n}$ is bijective on $m$-simplices for $m < n$ (see Definition 3.5.4.14). This is clear, since the morphisms $v_{n-1}^{\circ }$ and $u_{n}: X \rightarrow \pi _{\leq n}(X)$ are bijective on $m$-simplices.

Assertion $(3)$ follows by combining $(2)$ with Proposition 3.5.4.26. It remains to prove $(4)$. Fix a vertex $x \in X$, and let us abuse notation by identifying $x$ with its images in $\pi _{\leq n}(X)$ and $\operatorname{cosk}_{n-1}^{\circ }(X)$. Let $Y$ denote the fiber $f_{n}^{-1} \{ x\} $. It follows from $(3)$ that $Y$ is a Kan complex. Applying Remark 3.5.5.6, we see that $Y$ is an $n$-groupoid. Since $f_{n}$ is bijective on $m$-simplices for $m < n$, the simplicial set $Y$ has a single $m$-simplex for $m < n$. Applying Proposition 3.5.5.16, we obtain an isomorphism $Y \xrightarrow {\sim } \mathrm{K}(G,n)$ where $G$ is a set if $n=0$, a group if $n=1$, and an abelian group if $n \geq 2$. To complete the proof, it suffices to observe that the tautological maps

\[ G \simeq \pi _{n}(Y,x) \rightarrow \pi _{n}( \pi _{\leq n}(X), x) \xleftarrow \pi _{n}(X,x) \]

are isomorphisms: for the map on the right, this follows from Example 3.5.7.25, and for the map on the left it follows from the long exact sequence of Theorem 3.2.6.1. $\square$

Remark 3.5.8.8. Let $X$ be a Kan complex. Then the morphisms $f_{n}: \pi _{\leq n}(X) \rightarrow \operatorname{cosk}_{n-1}^{\circ }(X)$ of Proposition 3.5.8.7 fit into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [r] & \operatorname{cosk}_{3}^{\circ }(X) \ar [r] \ar [d] & \operatorname{cosk}_{2}^{\circ }(X) \ar [r] \ar [d] & \operatorname{cosk}_{1}^{\circ }(X) \ar [r] \ar [d] & \operatorname{cosk}_{0}^{\circ }(X) \ar [d] \\ \cdots \ar [r] \ar [ur] & \pi _{\leq 3}(X) \ar [r] \ar [ur]^{f_3} & \pi _{\leq 2}(X) \ar [r] \ar [ur]^{f_2} & \pi _{\leq 1}(X) \ar [ur]^{f_1} \ar [r] & \pi _{\leq 0}(X), } \]

which intertwines the intertwine the canonical Postnikov tower of Example 3.5.8.2 with the weakly coskeletal tower of Example 3.5.8.5.

Corollary 3.5.8.9. Let $X$ be a Kan complex. Then the transition maps in the canonical Postnikov tower

\[ \cdots \rightarrow \pi _{\leq 3}(X) \rightarrow \pi _{\leq 2}(X) \rightarrow \pi _{\leq 1}(X) \rightarrow \pi _{\leq 0}(X) \]

are Kan fibrations. Moreover, for every vertex $x \in X$ and every integer $n > 0$, there is a canonical homotopy equivalence

\[ \mathrm{K}(G,n) \rightarrow \{ x\} \times _{ \pi _{\leq n-1}(X) } \pi _{\leq n}(X), \]

for $G = \pi _{n}(X,x)$.

Proof. For $n > 0$, the transition map $\pi _{\leq n}(X) \rightarrow \pi _{\leq n-1}(X)$ factors as a composition

\[ \pi _{\leq n}(X) \xrightarrow {f_ n} \operatorname{cosk}_{n-1}^{\circ }(X) \xrightarrow {g} \pi _{\leq n-1}(X), \]

where $f_{n}$ is the Kan fibration of Proposition 3.5.8.7 and $g$ is the trivial Kan fibration of Corollary 3.5.6.14. We therefore obtain a pullback diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \mathrm{K}(G,n) \ar [r] \ar [d] & \{ x\} \times _{ \pi _{\leq n-1}(X) } \pi _{\leq n}(X) \ar [d] \\ \{ x\} \ar [r] & \{ x\} \times _{ \pi _{\leq n-1}(X) } \operatorname{cosk}_{n-1}^{\circ }(X) } \]

where the vertical maps are Kan fibrations and the lower right corner is contractible. In particular, the lower horizontal map is a homotopy equivalence. Applying Corollary 3.4.1.5, we deduce that the upper horizontal map is also a homotopy equivalence. $\square$

Variant 3.5.8.10. Let $X$ be a Kan complex. Then the transition maps in the weakly coskeletal tower

\[ \cdots \rightarrow \operatorname{cosk}_{3}^{\circ }(X) \rightarrow \operatorname{cosk}_{2}^{\circ }(X) \rightarrow \operatorname{cosk}_{1}^{\circ }(X) \rightarrow \operatorname{cosk}_{0}^{\circ }(X) \]

are Kan fibrations (whose fibers are homotopy equivalent to Eilenberg-MacLane spaces).

Warning 3.5.8.11. Let $X$ be a Kan complex. Then the transition maps in the coskeletal tower

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

are generally not Kan fibrations. See Warning 3.5.4.27.