Warning 11.4.0.11. Throughout most of this book, we will often employ the following conventions:
If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, then we let $\operatorname{\mathcal{C}}[W^{-1}]$ denote a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Remark 6.3.2.2).
If $\operatorname{\mathcal{C}}$ is an ordinary category, we abuse terminology by identifying $\operatorname{\mathcal{C}}$ with the associated $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
Beware that these conventions are in conflict with one another. If $W$ is a collection of morphisms in an ordinary category $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}[W^{-1}]$ denotes the strict localization of Remark 6.3.0.2, then the $\infty $-categorical localization $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})[W^{-1}]$ is generally not equivalent to the nerve $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}[W^{-1}] )$. To avoid confusion, we will henceforth use the notation $\operatorname{\mathcal{C}}[W^{-1}]$ only for the $\infty $-categorical notion of localization, unless otherwise specified.