Definition 4.6.3.3. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. A categorical mapping cylinder for $B$ relative to $A$ is a simplicial set $\overline{B}$ equipped with a morphism $\pi : \overline{B} \rightarrow B$ together with a pair of sections $s_0, s_1: B \rightarrow \overline{B}$ having the following properties:
The morphism $\pi : \overline{B} \rightarrow B$ is a categorical equivalence of simplicial sets.
The morphisms $s_0, s_1: B \rightarrow \overline{B}$ satisfy $s_0 \circ i = s_1 \circ i$, and the induced map $(s_0, s_1): (B \coprod _{A} B) \rightarrow \overline{B}$ is a monomorphism.
If these conditions are satisfied in the special case $A = \emptyset $, we will simply refer to $\overline{B}$ (together with the morphisms $\pi $, $s_0$, and $s_1$) as a categorical mapping cylinder for $B$.