Notation 4.5.8.3 (The Blunt Join). Let $X$ and $Y$ be simplicial sets. We let $X \diamond Y$ denote the simplicial set given by the iterated pushout

\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times \Delta ^1 \times Y) \coprod _{ ( X \times \{ 1\} \times Y) } Y, \]

so that we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \times \operatorname{\partial \Delta }^1 \times Y \ar [r] \ar [d]^{\pi _{X} \coprod \pi _{Y}} & X \times \Delta ^1 \times Y \ar [d] \\ ( X \times \{ 0\} ) \coprod ( \{ 1\} \times Y ) \ar [r] & X \diamond Y. } \]

We will refer $X \diamond Y$ as the *blunt join* of $X$ and $Y$. The commutative diagram (4.44) determines a morphism of simplicial sets $c_{X,Y}: X \diamond Y \rightarrow X \star Y$, which we will refer to as the *comparison map*.