Corollary 6.3.2.9 (05ZL)

Corollary 6.3.2.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, and form a pushout diagram of simplicial sets

6.14

\begin{equation} \begin{gathered}\label{equation:make-general-localization} \xymatrix@C =50pt@R=50pt{ \coprod _{w \in W} \Delta ^1 \ar [r] \ar [d]^{\coprod _{w \in W} e} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \coprod _{w \in W} Q \ar [r] & \operatorname{\mathcal{D}}, } \end{gathered} \end{equation}

where $e: \Delta ^1 \rightarrow Q$ is the morphism described in Corollary 6.3.2.8. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

Proof. Note that $e$ is a monomorphism of simplicial sets: that is, it corresponds to an edge of $Q$ where the source and target are distinct. It follows that the diagram (6.14) is a categorical pushout square (Example 4.5.4.12). By virtue of Proposition 6.3.2.4, it will suffice to show that the left vertical map of (6.14) exhibits $\operatorname{\mathcal{D}}_0 = \coprod _{w \in W} Q$ as a localization of $\operatorname{\mathcal{C}}_0 = \coprod _{w \in W} \Delta ^1$ with respect to the collection of nondegenerate edges of $\operatorname{\mathcal{C}}_0$. This follows by combining Corollary 6.3.2.8 with Remark 6.3.1.15. $\square$