ScalingStacks

Proof. For the special case $i=0$, a more general version of this statement was proven by Rezk [34, Proposition 6.6] and forms one of the cornerstone results of that work. Our current proof builds on Rezk’s ideas.

The fundamental argument is to construct a category $\mathcal{Q}$ along with a functorial assignment of commuting squares

for each $\alpha\in\mathcal{Q}$. This assignment is required to satisfy a host of conditions.

First, each of the functors $A,B:\mathcal{Q}\to\pre(\Theta_{n})$ is required to factor through $\tau_{\leq 0}\pre(\Theta_{n})$, the category of 0-truncated objects. The 0-truncated objects of $\pre(\Theta_{n})$ consist precisely of those presheaves of spaces taking values in the homotopically discrete spaces. There is no harm regarding such objects simply as ordinary set-valued presheaves, and we will do so freely.

Second, we require that for each $\alpha\in\mathcal{Q}$ the natural morphism $A_{\alpha}\to B_{\alpha}$ is in the class $T_{\Theta_{n}}$. As $T_{\Theta_{n}}$ is saturated, this second condition implies that the natural map $\colim_{\mathcal{Q}}A\to\colim_{\mathcal{Q}}B$ is also in $T_{\Theta_{n}}$, where these colimits are taken in the $\infty$-category $\pre(\Theta_{n})$ (hence are equivalently homotopy colimits for a levelwise model structure on simplical preseheaves, see Rk. 13.1).

Third and last, we require that the natural maps $\colim_{\mathcal{Q}}A\to U\times_{C_{i}}H$ and $\colim_{\mathcal{Q}}B\to V\times_{C_{i}}H$ are equivalences in $\pre(\Theta_{n})$ (i.e., levelwise weak equivalences of space-valued presheaves). If all of the above properties hold, then we obtain a natural commuting square

in which the indicated morphisms are in the class $T_{\Theta_{n}}$. As this class is saturated it follows that $U\times_{C_{i}}H\to V\times_{C_{i}}H$ is also in this class. Thus if such a $\mathcal{Q}$ and associated functors can be produced, we will have completed the proof.

At this point we deviate from Rezk’s treatment. Specifically our category $\mathcal{Q}$ and associated functors will differ from his. We will focus on the more complicated case $i>0$, and leave the necessary simplifications in the case $i=0$ to the reader (or simply refer the reader to [34, Proposition 6.6] ).

Under the assumptions of the statement of the lemma we have the following identifications of presheaves:

	$\displaystyle U$	$\displaystyle=j({\{0,\dots,k\}};o_{1},\dots,o_{k})\cup^{j({\{k\}})}j({\{k,k+1,\dots,m\}};o_{k+1},\dots,o_{m})$
	$\displaystyle V$	$\displaystyle=j({[m]};o_{1},\dots,o_{m})$
	$\displaystyle H$	$\displaystyle=j([n];u_{1},\dots,u_{n})$

where $0\leq k\leq m$ and $o_{\alpha},u_{\beta}\in\Theta_{n-1}$ are given. If $i>0$, then the $i$-cell is the representable presheaf $j([1];C_{i-1})$. A nondegenerate map $V\to C_{i}$ includes a nondegenerate map $f:[m]\to[1]$, and likewise a nondegenerate map $H\to C_{i}$ includes a nondegenerate map $g:[n]\to[1]$. Let $m^{\prime}$ be the fiber over $0\in[1]$, and let $m^{\prime\prime}$ be the fiber over $1$. Then $[m]=[m^{\prime}]\cdot[m^{\prime\prime}]$ is the ordered concatenation of $[m^{\prime}]$ and $[m^{\prime\prime}]$. Similarly $[n]=[n^{\prime}]\cdot[n^{\prime\prime}]$ is the ordered concatenation of the preimages of $0$ and $1$ under $g$.

Let $\delta=(\delta^{\prime},\delta^{\prime\prime}):[p]\to[m]\times_{[1]}[n]$ be a map which is an inclusion. There is a unique $-1\leq r\leq p$ such that under the composite $[p]\to[m]\times_{[1]}[n]\to[1]$, an element $s$ maps to $0$ if and only if $s\leq r$ (hence maps to $1$ if and only if $s>r$). Associated to $\delta$ we have a subobject $C_{\delta}$ of $V\times_{C_{i}}H$, of the form $C_{\delta}=\sigma_{!}^{[p]}(c_{1},\dots,c_{p})$, where $c_{\ell}$ is given by the following formula:

$$\prod_{\delta^{\prime}(\ell-1)<\alpha\leq\delta^{\prime}(\ell)}o_{\alpha}\times\prod_{\delta^{\prime\prime}(\ell-1)<\beta\leq\delta^{\prime\prime}(\ell)}u_{\beta}$$

if $\ell-1\neq r$, and if $\ell-1=r$ by

$$\left(\prod_{\alpha}o_{\alpha}\right)\times\left(\prod_{\beta}u_{\beta}\right)\times\left(o_{m^{\prime}}\times_{C_{i}}u_{n^{\prime}}\right)\times\left(\prod_{\lambda}o_{\lambda}\right)\times\left(\prod_{\epsilon}u_{\epsilon}\right)$$

where the indices range over all $\delta^{\prime}(\ell-1)<\alpha<m^{\prime}$, $\delta^{\prime\prime}(\ell-1)<\beta<n^{\prime}$, $m^{\prime}<\lambda\leq\delta^{\prime}(\ell)$, and $m^{\prime}<\beta\leq\epsilon^{\prime\prime}(\ell-1)$. We have found the graphical image in Figure 1 to be especially useful in understanding the combinatorics of these subobjects.

As subobjects of $V\times_{C_{i}}H$, the $C_{\delta}$ are naturally arranged into a poset. Let $W$ denote the disjoint union of all the maximal elements of this poset. Let $B_{\bullet}$ denote the simplicial Čech nerve associated to the morphism $W\to V\times_{C_{i}}H$. Each layer of $B_{\bullet}$ consists of a disjoint union of certain $C_{\delta}$. The map $W\to V\times_{C_{i}}H$ is a surjective map of set-valued presheaves. It follows that it is also an effective epimorphism in the $\infty$-topos $\pre(\Theta_{n})$, and hence [28, Corollary 6.2.3.5] the (homotopy) colimit of the simplcial diagram $B_{\bullet}$ is equivalent to $V\times_{C_{i}}H$. We set $\mathcal{Q}=\Delta$ and $B=B_{\bullet}$.

We define $A_{\bullet}$ to be the fiber product of $B_{\bullet}$ with $U\times_{C_{i}}H$ over $V\times_{C_{i}}H$. Because colimits in $\infty$-topoi are universal, we have

	$\displaystyle\colim_{\mathcal{Q}}A$	$\displaystyle\simeq\colim_{\mathcal{Q}}\left(B\times_{(V\times_{C_{i}}H)}(U\times_{C_{i}}H)\right)$
		$\displaystyle\simeq\left(\colim_{\Delta}B_{\bullet}\right)\times_{(V\times_{C_{i}}H)}(U\times_{C_{i}}H)$
		$\displaystyle\simeq U\times_{C_{i}}H.$

Thus all that remains is to show that the natural transformation $A_{\bullet}\to B_{\bullet}$ is levelwise in $T_{\Theta_{n}}$.

As each layer of $B_{\bullet}$ is a disjoint union of certain $C_{\delta}$, it is sufficient to show that the map

$$C_{\delta}\times_{(V\times_{C_{i}}H)}(U\times_{C_{i}}H)\to C_{\delta}$$

is in $T_{\Theta}$ for each $C_{\delta}$. As in the previous construction, there exist unique $0\leq r\leq s\leq p$ such that $\delta^{\prime}(t)<k$ if and only if $t<r$, and $k<\delta^{\prime}(t)$ if and only if $s<t$. The interval $\{r,\dots,s\}\subset[p]$ is precisely the preimage of $\{k\}$ under $\delta^{\prime}$. We then have

	$\displaystyle C_{\delta}$	$\displaystyle\times_{(V\times_{C_{i}}H)}(U\times_{C_{i}}H)$
		$\displaystyle\cong\sigma_{!}^{\{0,\dots,s\}}(b_{1},\dots,b_{s})\cup^{\sigma^{\{r,\dots,s\}}_{!}(b_{r+1},\dots,b_{s})}\sigma_{!}^{\{r,r+1,\dots,p\}}(b_{r+1},\dots,b_{p})$

and so the desired result follows from Lemma 13.5. ∎