Corollary 6.3.2.5 (05ZG)

Corollary 6.3.2.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{D}}$ be the quotient of $\operatorname{\mathcal{C}}$ obtained by collapsing each edge $w \in W$, so that we have a pushout diagram of simplicial sets

6.13

\begin{equation} \begin{gathered}\label{equation:easy-localization} \xymatrix@C =50pt@R=50pt{ \coprod _{w \in W} \Delta ^1 \ar [r]^{\iota } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \coprod _{w \in W} \Delta ^0 \ar [r] & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}

If $\iota $ is a monomorphism of simplicial sets, then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

Proof. The assumption that $\iota $ is a monomorphism guarantees that the diagram (6.13) 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 (6.13) exhibits $\operatorname{\mathcal{D}}_0 = \coprod _{w \in W} \Delta ^0$ 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 Example 6.3.1.14 with Remark 6.3.1.15. $\square$