Kerodon

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

Example 4.6.4.6. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets, and let $Q$ be a Kan complex containing vertices $x_0, x_1 \in Q$ with $x_0 \neq x_1$. Set $\overline{B} = A \coprod _{ (Q \times A) } (Q \times B)$. The commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ Q \times A \ar [r] \ar [d] & Q \times B \ar [d] \\ A \ar [r]^-{i} & B } \]

is a categorical pushout square (since the vertical maps are categorical equivalences), and therefore induces a categorical equivalence $\overline{\pi }: \overline{B} \rightarrow B$ (Proposition 4.5.3.7). Let $s_0: B \rightarrow \overline{B}$ be the section of $\pi $ given by the composition

\[ B \simeq \{ x_0 \} \times B \hookrightarrow Q \times B \rightarrow \overline{B}, \]

and define $s_1: B \rightarrow \overline{B}$ similarly. Then the quadruple $(\overline{B}, \pi , s_0, s_1)$ is a categorical mapping cylinder of $B$ relative to $A$.