Construction 5.2.3.1 (0242): The Relative Join

Construction 5.2.3.1 (The Relative Join). Let $\operatorname{\mathcal{E}}$ be a simplicial set. By virtue of Remark 4.3.3.23, there is a unique morphism of simplicial sets $\rho : \Delta ^1 \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \operatorname{\mathcal{E}}\ar [r] \ar [d]^-{\operatorname{id}_{\operatorname{\mathcal{E}}}} & \Delta ^1 \times \operatorname{\mathcal{E}}\ar [d]^-{\rho } & \{ 1\} \times \operatorname{\mathcal{E}}\ar [l] \ar [d]^-{\operatorname{id}_{\operatorname{\mathcal{E}}}} \\ \operatorname{\mathcal{E}}\star \, \, \emptyset \ar [r] & \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}& \emptyset \star \operatorname{\mathcal{E}}\ar [l] } \]

is commutative.

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. We let $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the fiber product $(\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}) \times _{ (\operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}})} (\Delta ^1 \times \operatorname{\mathcal{E}})$, so that we have a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d] & \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{E}}\ar [r]^-{\rho } & \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}. } \]

We will refer to $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the join of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ relative to $\operatorname{\mathcal{E}}$.