Exercise 3.1.6.21. Let $G$ be the directed graph depicted in the diagram

\[ \xymatrix@R =50pt@C=50pt{ 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & 3 \ar [r] & 4 \ar [r] & \cdots } \]

and let $G_{}$ denote the associated $1$-dimensional simplicial set (see Warning 1.2.1.28). Show that the projection map $G_{} \rightarrow \Delta ^{0}$ is a weak homotopy equivalence, but not a homotopy equivalence.