Example 5.3.2.13 (The Mapping Cylinder). Let $f: X \rightarrow Y$ be a morphism of simplicial sets, which we identify with a diagram $\mathscr {F}: [1] \rightarrow \operatorname{Set_{\Delta }}$. We will denote the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ by $ \underset { \longrightarrow }{\mathrm{holim}}( f: X \rightarrow Y)$ and refer to it as the mapping cylinder of the morphism $f$. Applying Remark 5.3.2.12, we obtain an isomorphism of simplicial sets
\[ \underset { \longrightarrow }{\mathrm{holim}}(f: X \rightarrow Y) \simeq (\Delta ^1 \times X) {\coprod }_{ ( \{ 1\} \times X) } Y; \]
that is, the mapping cylinder $ \underset { \longrightarrow }{\mathrm{holim}}(f: X \rightarrow Y)$ can be identified with the homotopy pushout $X {\coprod }_{X}^{\mathrm{h}} Y$ of Construction 3.4.2.2.