Kerodon

Notation 9.2.1.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a localizing collection of morphisms of $\operatorname{\mathcal{C}}$. We will often write $\operatorname{\mathcal{C}}[W^{-1}]$ for the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. By virtue of Proposition 9.2.1.18, this is consistent with Remark 6.3.2.2: that is, we can regard $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. This convention is very convenient, since the full subcategory of $W$-local objects is uniquely determined by $\operatorname{\mathcal{C}}$ and $W$. However, it has the potential to create confusion in some situations: see Warning 9.2.1.23 below.