Example 5.3.4.16 (0377)

Example 5.3.4.16. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, which we identify with a diagram $\mathscr {F}: [1] \rightarrow \operatorname{Set_{\Delta }}$. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the mapping cylinder $(\Delta ^1 \times X) {\coprod }_{ (\{ 1\} \times X) } Y$ (Example 5.3.2.13), and the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}([1])$ can be identified with the relative join $X \star _{Y} Y$ (Example 5.3.3.13). Under these identifications, Construction 5.3.4.11 corresponds to a morphism of simplicial sets

\[ \lambda _{t}: (\Delta ^1 \times X) {\coprod }_{ (\{ 1\} \times X) } Y \rightarrow X \star _{Y} Y. \]

Unwinding the definitions, we see that this map classifies the commutative diagram

5.27

\begin{equation} \begin{gathered}\label{equation:comparison-map-over-edge} \xymatrix@R =50pt@C=50pt{ \emptyset \star _{X} X \ar [r] \ar [d] & \emptyset \star _{Y} Y \ar [d] \\ X \star _{X} X \ar [r] & X \star _{Y} Y. } \end{gathered} \end{equation}

In particular, the morphism $\lambda _{t}$ is an isomorphism if and only if (5.27) is a pushout square of simplicial sets.