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Example (Homotopy Quotients). Let $G$ be a group and let $BG$ denote the associated groupoid (consisting of a single object with automorphism group $G$). Let $X$ be a simplicial set equipped with an action of $G$, which we identify with a functor $\mathscr {F}: BG \rightarrow \operatorname{Set_{\Delta }}$. We will denote the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ by $X_{\mathrm{h}G}$, and refer to it as the homotopy quotient of $X$ by the action of $G$.