ScalingStacks

Definition 2.9. Given an $\infty$-category $\mathcal{C}$ and an integer $d\geq-2$, we define the $d$-homotopy category of $\mathcal{C}$ to be $h_{d}\mathcal{C}$ of 2.8 when $d\geq 1$ and

1. (1)

   For $d=-2$ we set $h_{-2}\mathcal{C}=\Delta^{0}$.

2. (2)

   For $d=-1$ we set $h_{-1}\mathcal{C}=\begin{cases}\varnothing&\mathcal{C}=\varnothing\\ \Delta^{0}&\mathcal{C}\neq\varnothing\end{cases}$ with the unique map $\theta_{-1}\colon\mathcal{C}\to h_{-1}\mathcal{C}$.

3. (3)

   For $d=0$, we first define a pre-ordered set $\tilde{h}_{0}\mathcal{C}$ with the same objects as $\mathcal{C}$ and the relation $x\leq y$ if and only if $\operatorname{Map}_{\mathcal{C}}\left(X,Y\right)\neq\varnothing$. Then we define $h_{0}\mathcal{C}$ to be the nerve of the poset obtained from $\tilde{h}_{0}\mathcal{C}$ by identifying isomorphic objects. There is a canonical map $\theta_{0}\colon\mathcal{C}\to h_{0}\mathcal{C}$ defined as the composition of $\theta_{1}\colon\mathcal{C}\to h_{1}\mathcal{C}$ with the nerve of the functor that takes each object in the homotopy category $h_{1}\mathcal{C}$ to its class in $h_{0}\mathcal{C}$ (with the unique definition on morphisms).