Example 6.3.2.6 (Contracting an Edge). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $w: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{C}}$ where $X \neq Y$. Let $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}/ \Delta ^1$ be the simplicial set obtained from $\operatorname{\mathcal{C}}$ by collapsing the edge $w$. Then the quotient map $\operatorname{\mathcal{C}}\twoheadrightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W = \{ w\} $.
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