Kerodon

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

Example 3.4.1.13. Let $X$ be a Kan complex containing a vertex $x \in X$, let $\Omega X$ denote the loop space $\{ x\} \times ^{\mathrm{h}}_{X} \{ x\} $, and let $P$ denote the path space $X \times _{X}^{\mathrm{h}} \{ x\} $, and let $\iota : \Omega X \hookrightarrow P$ be the inclusion map. We then have a pullback diagram of Kan complexes

3.46
\begin{equation} \begin{gathered}\label{equation:path-space-counterexample} \xymatrix@R =50pt@C=50pt{ \Omega X \ar [r]^-{\iota } \ar [d] & P \ar [d]^{\operatorname{ev}_0} \\ \{ x\} \ar [r] & X, } \end{gathered} \end{equation}

where $\operatorname{ev}_0$ is given by evaluation at the vertex $0 \in \Delta ^1$. Since $\operatorname{ev}_0$ is a Kan fibration, the diagram (3.46) is also a homotopy pullback square (Example 3.4.1.3). Note that the Kan complex $P$ is contractible, so that $\iota $ is homotopic to the constant map $\iota ': \Omega X \rightarrow P$ carrying $\Omega X$ to the constant path $\operatorname{id}_{x}$. However, the commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \Omega X \ar [r]^-{\iota '} \ar [d] & P \ar [d]^{\operatorname{ev}_0} \\ \{ x\} \ar [r] & X } \]

is never a homotopy pullback square unless the Kan complex $\Omega X$ is contractible (again by Example 3.4.1.3).