Kerodon

Proposition 6.3.2.11. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. Then there exists an $\infty $-category $\operatorname{\mathcal{C}}[W^{-1}]$ and a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Moreover, we can arrange that the assignment $(\operatorname{\mathcal{C}},W) \mapsto ( \operatorname{\mathcal{C}}[W^{-1}], F )$ is functorial and commutes with filtered colimits.