Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.2.6.7. Let $X$ be a Kan complex. The following conditions are equivalent:

$(1)$

The projection map $X \rightarrow \Delta ^{0}$ is a trivial Kan fibration.

$(2)$

The Kan complex $X$ is contractible: that is, the projection map $X \rightarrow \Delta ^{0}$ is a homotopy equivalence.

$(3)$

The Kan complex $X$ is nonempty. Moreover, for each vertex $x \in X$ and each $n \geq 0$, the set $\pi _{n}(X,x)$ has a single element.

$(4)$

The Kan complex $X$ is connected. Moreover, there exists a vertex $x \in X$ such that the homotopy groups $\pi _{n}(X,x)$ are trivial for $n \geq 1$.

Proof. The implication $(1) \Rightarrow (2)$ is a special case of Proposition 3.1.5.9, the implication $(2) \Rightarrow (3)$ follows from Corollary 3.2.6.3, and the implication $(3) \Rightarrow (4)$ is immediate. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that $X$ is connected, and fix a vertex $x \in X$ for which the homotopy groups $\pi _{n}(X,x)$ vanish for $n \geq 1$. We first prove the following:

$(\ast )$

Let $f: B \rightarrow X$ be a morphism of simplicial sets, let $A \subseteq B$ be a simplicial subset, and let $h: \Delta ^{1} \times A \rightarrow X$ be a homotopy from $f|_{A}$ to the constant map $A \rightarrow \{ x\} \subseteq X$. Then $h$ can be extended to a homotopy $\overline{h}: \Delta ^1 \times B \rightarrow X$ from $f$ to the constant map $B \rightarrow \{ x \} \subseteq X$.

To prove $(\ast )$, we may assume without loss of generality that $B = \Delta ^ n$ and $A = \operatorname{\partial \Delta }^ n$ for some $n \geq 0$. Since $X$ is a Kan complex, we can extend $h$ to a homotopy $\overline{h}': \Delta ^1 \times \Delta ^ n \rightarrow X$ from $f$ to some other map $f': \Delta ^{n} \rightarrow X$ for which $f'|_{ \operatorname{\partial \Delta }^{n} }$ is the constant map taking the value $x$ (Remark 3.1.4.3). Replacing $f$ by $f'$, we can reduce to the case where $f|_{ \operatorname{\partial \Delta }^{n} }$ and $h: \Delta ^1 \times \operatorname{\partial \Delta }^ n \rightarrow X$ are the constant maps taking the value $x$. In this case, the existence of the desired extension follows from the vanishing of the homotopy group $\pi _{n}(X,x)$ (or from the connectedness of $X$, in the special case $n=0$).

We now prove that the map $X \rightarrow \Delta ^{0}$ is a trivial Kan fibration. Fix an integer $n \geq 0$ and a morphism of simplicial sets $f: \operatorname{\partial \Delta }^{n} \rightarrow X$; we wish to show that $f$ can be extended to an $n$-simplex of $X$. Applying $(\ast )$ in the case $B = \operatorname{\partial \Delta }^{n}$ and $A = \emptyset $, we conclude that there exists a homotopy $h: \Delta ^1 \times \operatorname{\partial \Delta }^ n \rightarrow X$ from $f$ to the constant map $\operatorname{\partial \Delta }^{n} \rightarrow \{ x\} \subseteq X$. Because $X$ is a Kan complex, we can extend $h$ to a map $\overline{h}: \Delta ^1 \times \Delta ^ n \rightarrow X$ for which $\overline{h}|_{\{ 1\} \times \Delta ^ n}$ is the constant map taking the value $x$ (Remark 3.1.4.3). The restriction $\overline{h}|_{ \{ 0\} \times \Delta ^ n }$ then provides the desired extension of $f$. $\square$