ScalingStacks

Definition 4.2.2. (T.5.5.6.1) For $d\geq-2$, a map $f\colon X\to Y$ in an $\infty$-category $\mathcal{C}$ is called $d$-truncated, if for every $Z\in\mathcal{C}$ the induced map

$$\operatorname{Map}\left(Z,X\right)\to\operatorname{Map}\left(Z,Y\right)$$

is a $d$-truncated map of spaces. An object $X$ is $d$-truncated, if the map $X\to\text{pt}_{\mathcal{C}}$ is $d$-truncated. We denote by $\tau_{\leq d}\mathcal{C}$ the full subcategory of $\mathcal{C}$ spanned by the $d$-truncated objects. When $\mathcal{C}$ is presentable, by T.5.5.6.21 the $\infty$-category $\tau_{\leq d}\mathcal{C}$ is itself presentable and by T.5.5.6.18, the inclusion $\tau_{\leq d}\mathcal{C}\hookrightarrow\mathcal{C}$ has a left adjoint $\tau_{\leq d}^{\mathcal{C}}\colon\mathcal{C}\to\tau_{\leq d}\mathcal{C}$.