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
\[ G \simeq \pi _{n}(Y,x) \rightarrow \pi _{n}( \pi _{\leq n}(X), x) \leftarrow \pi _{n}(X,x) \]
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.
$\square$