ScalingStacks

Definition 2.2. Let $d\geq-1$ be an integer. An essentially $d$-category is an $\infty$-category $\mathcal{C}$ such that for all $X,Y\in\mathcal{C}$, the mapping space $\operatorname{Map}_{\mathcal{C}}\left(X,Y\right)$ is $\left(d-1\right)$-truncated. We denote by $\mathbf{Cat}_{d}$ the full subcategory of $\mathbf{Cat}_{\infty}$ spanned by essentially $d$ -categories.