ScalingStacks

Proof. If $n=0$, there is nothing to prove. If $n$ is positive, then let us suppose that we have written $\CSS_{n-1}$ as a localization $T_{\Delta^{\!\times n-1}}^{-1}\pre(\Delta^{\!\times n-1})$ for some strongly saturated class $T_{\Delta^{\!\times n-1}}$ of small generation. Denote by

$$\boxtimes\colon\pre(\Delta)\times\pre(\Delta^{\!\times n-1})\to\pre(\Delta^{\!\times n})$$

the essentially unique functor that carries pairs of the form $(j[k],j(\mathbf{m}))$ to $j([k],\mathbf{m})$ and preserves colimits separately in each variable. Now let $T_{\Delta^{\!\times n}}$ be the strongly saturated class generated by the class $T$ above along with the set

$$\{j[k]\boxtimes U\to j[k]\boxtimes V\ |\ [U\to V]\in T_{\Delta^{\!\times n-1}}\}.$$

Now $\CSS(\Delta^{\!\times n})$ coincides with $T_{\Delta^{\!\times n}}^{-1}\pre(\Delta^{\!\times n})$. ∎