Lemma 4.5.8.11. The comparison map $c_{ \Delta ^1, \Delta ^0}: \Delta ^{1} \diamond \Delta ^0 \rightarrow \Delta ^{1} \star \Delta ^0$ is a categorical equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Unwinding the definitions, we can identify the blunt join $\Delta ^1 \diamond \Delta ^0$ with the simplicial set $(\Delta ^1 \times \Delta ^1) \coprod _{ \Delta ^{1} \times \{ 1\} } \Delta ^0$, which we represent informally by the diagram
\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [dr] \ar [d] & \bullet \ar@ {=}[d] \\ \bullet \ar [r] & \bullet . } \]
Let $X$ denote the simplicial set $\Delta ^{2} \coprod _{ \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) } \Delta ^0$ obtained from the standard $2$-simplex by collapsing the final edge to a point. We then have an inclusion map $\iota : X \hookrightarrow \Delta ^1 \diamond \Delta ^0$ (corresponding to the triangle in the upper right of the preceding diagram), which fits into a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{r} \ar [d] & \Delta ^1 \ar [d]^{u} \\ \Delta ^1 \diamond \Delta ^0 \ar [r]^-{c_{ \Delta ^1, \Delta ^0}} & \Delta ^1 \star \Delta ^0; } \]
here $u$ classifies to the “long edge” of the $2$-simplex $\Delta ^1 \star \Delta ^0 \simeq \Delta ^2$. Since the vertical maps are monomorphisms and $r$ is a categorical equivalence (see Example 4.5.3.16), it follows that $c_{\Delta ^1, \Delta ^0}$ is also a categorical equivalence (Remark 4.5.4.13). $\square$