Kerodon

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

Example 3.4.1.11. Let $S$ be a Kan complex containing a vertex $s \in S$, let $\Omega S$ denote the loop space $\operatorname{Fun}( \Delta ^1, S) \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, S)} \{ (s,s) \} $. Let $Q$ denote the path space $\operatorname{Fun}( \Delta ^1, S) \times _{ \operatorname{Fun}( \{ 1\} , S) } \{ s\} $, and let $\iota : \Omega S \hookrightarrow Q$ be the inclusion map. We then have a pullback diagram of Kan complexes $\sigma :$

\[ \xymatrix { \Omega S \ar [r]^{\iota } & Q \ar [d]^{\operatorname{ev}_0} \\ \{ s\} \ar [r] & S, } \]

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 $\sigma $ is also a homotopy pullback square (Example 3.4.1.5). Note that the Kan complex $Q$ is contractible, so that $\iota $ is homotopic to the constant map $\iota ': \Omega S \rightarrow Q$ carrying $\Omega S$ to the constant path $\operatorname{id}_{s}$. However, the commutative diagram of Kan complexes $\sigma ':$

\[ \xymatrix { \Omega S \ar [r]^{\iota '} & Q \ar [d]^{\operatorname{ev}_0} \\ \{ s\} \ar [r] & S } \]

is never a homotopy pullback square unless the Kan complex $\Omega S$ is contractible (again by Example 3.4.1.5).