ScalingStacks

Notation 6.5. Let $S_{00}$ consist of the union $A\cup B\cup C\cup D$ of the following four finite sets of maps of presheaves on $\Upsilon_{n}$:

$$A\mathrel{\mathop{:}}=\left\{\nu C_{i-1}\cup^{\nu(\partial C_{i-1})}\nu C_{i-1}\to\nu(\partial C_{i})\;|\;0\leq i\leq n-1\right\}$$

(when $i=0$, we interpret this as the empty presheaf mapping to the nerve of the empty $n$-category),

$$B\mathrel{\mathop{:}}=\left\{\nu C_{j}\cup^{\nu C_{i}}\nu C_{j}\to\nu(C_{j}\cup^{C_{i}}C_{j})\;|\;0\leq i<j\leq n\right\},$$

$$\begin{split}C\mathrel{\mathop{:}}=\Big\{\nu(C_{i+j}\cup^{C_{i}}C_{i+k})\cup^{\nu\sigma^{i+1}(C_{j-1}\times C_{k-1})}\nu(C_{i+k}&\cup^{C_{i}}C_{i+j})\to\nu(C_{i+j}\times_{C_{i}}C_{i+k})\\ &\Big|\;0\leq i\leq n\textrm{, }0<j,k\leq n-i\Big\},\end{split}$$

and, lastly,

$$D\mathrel{\mathop{:}}=\left\{\nu\sigma^{k}(\Delta^{3})\cup^{\nu\sigma^{k}(\Delta^{\{0,2\}}\sqcup\Delta^{\{1,3\}})}\nu\sigma^{k}(\Delta^{0}\sqcup\Delta^{0})\to\nu C_{k}\;|\;0\leq k\leq n\right\}.$$

Now let $S_{0}$ be the smallest class of morphisms $U\to V$ in $\Fun(\Upsilon_{n}^{\mathrm{op}},\set)$ that (a) is closed under isomorphism, (b) contains $S_{00}$, and (c) is closed under the operation $-\times_{C_{k}}N$ for any functor $V\to C_{k}$ and any $k$-correspondence $N\to C_{k}$ with $N\in\Upsilon_{n}$.