ScalingStacks

Proof. Lemma 8.4 shows that the restriction of $j^{\ast}$ to $\cat_{(\infty,n)}$ is fully faithful. Now to prove that $j^{\ast}$ is essentially surjective when restricted to $\cat_{(\infty,n)}$, it suffices to prove that $S^{-1}\pre(\Upsilon_{n})$ is generated under colimits by the cells. Since every object is a colimit of representables, it suffices to prove that $\Upsilon_{n}$ itself is generated under colimits in $S^{-1}\pre(\Upsilon_{n})$ by the cells. To prove this, we filter $\Upsilon_{n}$ in the following manner.

Let $\Upsilon_{n}^{(0)}=\mathbb{G}_{n}$ be the globular category of cells. For any $k\geq 1$, define $\Upsilon_{n}^{(k)}$ to be the full subcategory of $\Upsilon_{n}$ spanned by the set

$$\left\{X\in\Upsilon_{n}\;\middle|\;\begin{aligned} &\textrm{there exists a colimit diagram }f\colon K^{\rhd}\to S^{-1}\pre(\Upsilon_{n})\\ &\textrm{such that }f(+\infty)\simeq X\textrm{ and }f(K)\subset\Upsilon_{n}^{(k-1)}\end{aligned}\right\}.$$

That is, $\Upsilon_{n}^{(k)}\subset\Upsilon_{n}$ consists of colimits, formed in $S^{-1}\pre(\Upsilon_{n})$, of diagrams of objects of $\Upsilon_{n}^{(k-1)}$.

We claim that the collection $\{\Upsilon_{n}^{(k)}\}$ forms an exhaustive filtration of $\Upsilon_{n}$, so that we have $\cup_{k}\Upsilon_{n}^{(k)}=\Upsilon_{n}$. First we observe that the strongly saturated class $S$ contains the map

$$\sigma(i(o_{1}))\cup^{C_{0}}\sigma(i(o_{2}))\cup^{C_{0}}\cdots\cup^{C_{0}}\sigma(i(o_{m}))\to i([m];o_{1},\dots,o_{m})$$

and thus, by induction, the union $\cup_{k}\Upsilon_{n}^{(k)}$ contains $\Theta_{n}$.

It now suffices to show that this union is closed under fiber products over cells. Since colimits commute with fiber products over cells (both in $\gaunt_{n}$ and $S^{-1}\pre(\Upsilon_{n})$) it is sufficient to show that $C_{j}\times_{C_{i}}C_{k}$ is contained in the union for all $i,j,k\leq n$. The fiber products of cells were analyzed in detail in Remark 6.3 and the proof of Lemma 6.7, where it was shown that they can all be obtained from the cells by a finite number of the colimits provided by $S_{00}$. These are colimits in $\cat_{(\infty,n)}$, whence the result follows. ∎