ScalingStacks

Warning 3.1.5. 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. Both constructions will play a central role in the proof of the main result, and hopefully the distinction in notation and terminology will prevent confusion.

With these ideas in mind, one might hope that for an $\infty$-category $\mathcal{C}$, the condition of being an essentially $(d+1)$-category would coincide with the condition of begin a $d$-truncated object of the presentable $\infty$-category $\mathbf{Cat}_{\infty}$. This turns out to be false. More precisely, it can be shown that a $d$-truncated object of $\mathbf{Cat}_{\infty}$ is an essentially $(d+1)$-category and that an essentially $(d+1)$-category is a $(d+1)$-truncated object of $\mathbf{Cat}_{\infty}$, but neither of the converses hold (see [SY19, Remark 2.10]).