ScalingStacks

Proof. As $i_{!}$ commutes with colimits, to show that $i_{!}(T_{\Theta_{n}})\subseteq S$ it is sufficient to show this property for a subset that generates $T_{\Theta_{n}}$ under colimits. The maps in $\mathrm{Comp}_{\Theta_{n}}$ are clearly mapped into $S$. This leaves the maps $\mathrm{Segal}_{\Theta_{n}}$. We now write $S=S_{n}$ and induct on $n$. When $n=0$, one has $\Upsilon_{0}=\Theta_{0}=\mathrm{pt}$.

Assume that $i_{!}(T_{\Theta_{n-1}})\subseteq S_{n-1}$. The suspension functor $\sigma_{!}\colon\pre(\Upsilon_{n-1})\to\pre(\Upsilon_{n})$ preserves colimits and sends the generators $S_{n-1}$ into $S_{n}$. Hence the suspensions of maps in $i_{!}(T_{\Theta_{n-1}})$ are in $S_{n}$. Moreover, by construction the image under $i_{!}$ of the following map

$$j({\{0,1\}};C_{i})\cup^{j({\{1\}})}j({\{1,2\}};C_{i})\to j({\{0,1,2\}};C_{i},C_{i})$$

is in $S_{n}$ for all cells $C_{i}$. By induction, it follows that all the Segal generators are mapped into $S_{n}$ except possibly the following

(13.14.1)		$\displaystyle j({\{0,\dots,k\}};o_{1},\dots,o_{k})\cup^{j({\{k\}})}$	$\displaystyle j({\{k,\dots,m\}};o_{k+1},\dots,o_{m})$
		$\displaystyle\to j({\{0,\dots,m\}};o_{1},\dots,o_{m})$

where $o_{i}\in\Theta_{n-1}$. To show that $i_{!}$ maps the above morphism to a morphism in $S_{n}$, we observe that the above map may be rewritten as follows. The source may be written as

	$\displaystyle j({\{0,\dots,k\}};o_{1},\dots,o_{k})$	$\displaystyle\times_{j({\{k-1,k\}};C_{0})}\left[j({\{k-1,k\}};C_{0})\cup^{j({\{k\}})}j({\{k,k+1\}};C_{0})\right]$
		$\displaystyle\times_{j({\{k,k+1\}};C_{0})}j({\{k,k+1,m\}};o_{k+1},\dots,o_{m})$

while the target is

	$\displaystyle j({\{0,\dots,k\}};o_{1},\dots,o_{k})$	$\displaystyle\times_{j({\{k-1,k\}};C_{0})}j({\{k-1,k,k+1\}};C_{0},C_{0})$
		$\displaystyle\times_{j({\{k,k+1\}};C_{0})}j({\{k,k+1,m\}};o_{k+1},\dots,o_{m}).$

Schematically then, the map of (13.14.1) is of the form

$$A\times_{C_{1}}U\times_{C_{1}}B\to A\times_{C_{1}}V\times_{C_{1}}B$$

for $U\to V$ in $S$ and $A,B\in\Upsilon_{n}$. By property (C.2) of $\cat_{(\infty,n)}$ (cf. also Proposition 8.5) it follows that (13.14.1) lies in $S_{n}$ also. ∎