ScalingStacks

Warning 2.10. Note that an $\infty$-category $\mathcal{C}$ is an essentially $d$-category if and only if all objects of $\mathcal{C}$ are $\left(d-1\right)$-truncated in the sense of T.5.5.6.1. Hence, another way to associate an essentially $d$-category with an $\infty$-category $\mathcal{C}$ is to consider the full subcategory spanned by the $\left(d-1\right)$-truncated objects. For a presentable $\infty$-category, this is denoted by $\tau_{\leq d-1}\mathcal{C}$ in T.5.5.6.1 and called the $\left(d-1\right)$-truncation of $\mathcal{C}$. We warn the reader that the two essentially $d$-categories $h_{d}\mathcal{C}$ and $\tau_{\leq d-1}\mathcal{C}$ are usually very different. For example, when $\mathcal{C}=\mathcal{S}$ is the $\infty$-category of spaces, $h_{1}\mathcal{S}$ is the ordinary homotopy category of spaces, while $\tau_{\leq 0}\mathcal{S}$ is equivalent to the ordinary category of sets.