Kerodon

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Example 6.3.4.4 (Contracting an Edge). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $e$ be an edge of $\operatorname{\mathcal{C}}$ which corresponds to a monomorphism of simplicial sets $\Delta ^1 \hookrightarrow \operatorname{\mathcal{C}}$ (that is, the source and target of $e$ are distinct when regarded as vertices of $\operatorname{\mathcal{C}}$). Let $\operatorname{\mathcal{C}}'$ denote the simplicial set obtained from $\operatorname{\mathcal{C}}$ by collapsing the edge $e$, so that we have a pushout square of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Delta ^1 \ar [r]^{e} & \operatorname{\mathcal{C}}\ar [d]^{T} \\ \Delta ^{0} \ar [r] & \operatorname{\mathcal{C}}'. }$

Since the horizontal maps in this diagram are monomorphisms, it is also a categorical pushout square (Example 4.5.4.12). Combining Corollary 6.3.4.3 with Example 6.3.1.14, we see that $T$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to the singleton $W = \{ e \}$.