ScalingStacks

Definition 2.1. For $d\geq 0$, a space $X\in\mathcal{S}$ is called $d$-truncated if $\pi_{i}\left(X,x\right)=0$ for all $i>d$ and all $x\in X$. In addition, a space is called $\left(-2\right)$-truncated if and only if it is contractible and it is called $\left(-1\right)$-truncated if and only if it is either contractible or empty. We denote by $\mathcal{S}_{\leq d}$ the full subcategory of $\mathcal{S}$ spanned by the $d$-truncated spaces. The inclusion $\mathcal{S}_{\leq d}\hookrightarrow\mathcal{S}$ admits a left adjoint and we call the unit of the adjunction the $d$-truncation map.