Remark 13.12 (0MKN) · ScalingStacks

Remark 13.12. We note that the construction of a $\mathcal{Q}$ , $A$ , and $B$ demonstrably satisfying the above properties appears to be somewhat delicate. In the case $i=0$ the original published proof [34, Proposition 6.6] was incorrect, and a corrected proof has been supplied by Rezk in [35, Proposition 2.1].

It is possible to give an alternative proof of Lemma 13.11 which builds directly on Rezk’s proof in the case $i=0$ . Let $\mathcal{Q}$ be Rezk’s category $\mathcal{Q}_{m,n}$ as defined in [35], and following Rezk define functors $A^{(i=0)}$ and $B^{(i=0)}$ as the objects

	$\displaystyle A^{(i=0)}_{\delta}$	$\displaystyle:=V_{(\delta_{1},\delta_{2})^{-1}(M\times N)}(D_{1},\dots,D_{p}),\quad\text{and}$
	$\displaystyle B^{(i=0)}_{\delta}$	$\displaystyle:=V[p](D_{1},\dots,D_{p}),$

where we are also using the notation of [35]. Then these choices satisfy the requisite properties for the case $i=0$ , the most difficult being [35, Proposition 2.3].

For general $i$ , notice that we have inclusions of subobjects

	$\displaystyle V\times_{C_{i}}H$	$\displaystyle\subseteq V\times H,$
	$\displaystyle U\times_{C_{i}}H$	$\displaystyle\subseteq U\times H.$

We may define $A_{\delta}$ and $B_{\delta}$ as pullbacks

	$\displaystyle A_{\delta}$	$\displaystyle=A^{(i=0)}_{\delta}\times_{(U\times H)}(U\times_{C_{i}}H),\text{ and}$
	$\displaystyle B_{\delta}$	$\displaystyle=B^{(i=0)}_{\delta}\times_{(V\times H)}(V\times_{C_{i}}H).$

Since colimits in $\infty$ -topoi are universal we have

	$\displaystyle\colim_{\mathcal{Q}}A$	$\displaystyle\simeq\colim_{\mathcal{Q}}A^{(i=0)}\times_{(U\times H)}(U\times_{C_{i}}H)\simeq U\times_{C_{i}}H,$
	$\displaystyle\colim_{\mathcal{Q}}B$	$\displaystyle\simeq\colim_{\mathcal{Q}}B^{(i=0)}\times_{(V\times H)}(V\times_{C_{i}}H)\simeq V\times_{C_{i}}H,$

and so all that remains is to verify that $A_{\delta}\to B_{\delta}$ is indeed in $T_{\Theta_{n}}$ . This can be accomplished by explicitly computing $A_{\delta}$ and $B_{\delta}$ in terms of the functors $\sigma^{[\ell]}_{!}$ and invoking Lemma 13.5.