ScalingStacks

Notation 2.6. We may generalize the fourth example in the following manner. Suppose $X$ a strict $n$-category. We obtain a strict $(n+1)$-category $\sigma X$, the suspension of $X$, as follows. The set of objects of $\sigma X$ is the set $\{\top,\bot\}$, and one defines

$$\hom_{\sigma X}(x,y)\mathrel{\mathop{:}}=\begin{cases}C_{0}&\textrm{if }x=y;\\ X&\textrm{if }x=\bot\textrm{ and }y=\top;\\ \emptyset&\textrm{otherwise.}\end{cases}$$

There is a unique composition law that makes this into a strict $n$-category.

Observe that the $k$-fold suspension of the zero cell $C_{0}$ is now nothing more than the $k$-cell $\sigma^{k}(C_{0})=C_{k}$. Furthermore, the suspension functor preserves both pullback and pushout squares. Consequently, we have an isomorphism

$$\sigma(\emptyset)\cong C_{0}\sqcup C_{0}\cong\partial C_{1},$$

and therefore by induction we have

$$\sigma^{k}(\emptyset)\cong\sigma^{k-1}(C_{0}\cup^{\emptyset}C_{0})\cong C_{k-1}\cup^{\partial C_{k-1}}C_{k-1}\cong\partial C_{k}.$$

The canonical inclusion $\partial C_{k}\hookrightarrow C_{k-1}$ arises as the $k$-fold suspension of the unique functor $C_{0}\sqcup C_{0}\to C_{0}$.