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Definition 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 $B$.