ScalingStacks

Proof. We will proceed by induction on $n$, the base case $n=0$ being trivial. As the functor $j[m]\boxtimes(-)$ preserves colimits and sends the generators of $T_{\Delta^{\times n-1}}$ into $T_{\Delta^{\times n}}$, we have a containment $j[m]\boxtimes T_{\Delta^{\times n-1}}\subseteq T_{\Delta^{\times n}}$. Thus by induction the canonical map,

$$j[m]\boxtimes\mathbf{m}\to j[m]\boxtimes\delta^{*}_{n-1}\delta_{n-1}(\mathbf{m})$$

is in $T_{\Delta^{\times n}}$. In particular when $m=0$, the map $j[m]\boxtimes\mathbf{m}\to j[m]\boxtimes\mathbf{0}$ is a composite of maps in $\mathrm{Glob}_{\Delta^{\!\times n}}$, whence the map

$$j[0]\boxtimes\delta^{*}_{n-1}\delta_{n-1}(\mathbf{m})\to j[0]\boxtimes\mathbf{0}$$

is in $T_{\Delta^{\times n}}$.

We will first prove the lemma for objects of the form $j[1]\boxtimes\mathbf{m}\in\Delta\times\Delta^{\times n-1}\cong\Delta^{\times n}$. One may readily check that the following is a pushout square of presheaves of sets:

Moreover, as the topmost map is an inclusion of sets and pushouts in $\pre(\Delta^{\times n})$ are computed object-wise, this is also a (homotopy) pushout square in $\pre(\Delta^{\times n})$. As we just observed, the left-most map is in the strongly saturated $T_{\Delta^{\times n}}$, whence the right-most map is also in $T_{\Delta^{\times n}}$. It follows that the composite,

$$j[1]\boxtimes\mathbf{m}\to j[1]\boxtimes\delta^{*}_{n-1}\delta_{n-1}(\mathbf{m})\to\delta^{*}_{n}\delta_{n}(j[1]\boxtimes\mathbf{m})$$

is in $T_{\Delta^{\times n}}$.

To prove the general case, i.e., that the map $j[k]\boxtimes\mathbf{m}\to\delta^{*}_{n}\delta_{n}(j[k]\boxtimes\mathbf{m})$ is in $T_{\Delta^{\times n}}$, we induct on $k$. Assume the result holds when $k\leq m$. We will prove it for $k=m+1$. First consider the following commutative square:

The indicated maps are in $T_{\Delta^{\times n}}$; the topmost map is a generator and the lefttmost vertical map by induction. As $T_{\Delta^{\times n}}$ is saturated, the rightmost vertical map is in $T_{\Delta^{\times n}}$ if and only if the bottommost map is as well. Thus it suffices to prove that the natural map

$$\delta_{n}^{*}\delta_{n}(j[m]\boxtimes\mathbf{m})\cup^{j[0]\boxtimes\mathbf{0}}\delta_{n}^{*}\delta_{n}(j[1]\boxtimes\mathbf{m})\to\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m})$$

is in $T_{\Delta^{\times n}}$.

The Yoneda embedding is dense for any presheaf $\infty$-category and hence the object $\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m})$ may canonically be written as a colimit of representable presheaves. Let $\mathcal{D}=(\Delta^{\times n}\downarrow\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m}))$ denote the overcategory consisting of pairs $(j[p]\boxtimes\mathbf{p},\phi)$ where $\phi$ is a map

$$\phi:j[p]\boxtimes\mathbf{p}\to\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m}).$$

Let $B:\mathcal{D}\to\pre(\Delta^{\times n})$ denote the functor which forgets the map $\phi$. We have a canonical equivalence in $\pre(\Delta^{\times n})$:

$$\colim_{\mathcal{D}}B\simeq\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m}).$$

By adjunction, specifying a map $\phi:j[p]\boxtimes\mathbf{p}\to\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m})$ is equivalent to a specifying a map $\phi^{\prime}:\delta_{n}(j[p]\boxtimes\mathbf{p})\to\delta_{n}(j[m+1]\boxtimes\mathbf{m})$, i.e., a map in $\pre(\Theta_{n})$:

$$\phi^{\prime}:([p];\underbrace{\delta_{n-1}(\mathbf{p}),\dots,\delta_{n-1}(\mathbf{p})}_{p\text{ times}})\to([m+1];\underbrace{\delta_{n-1}(\mathbf{m}),\dots,\delta_{n-1}(\mathbf{m})}_{m+1\text{ times}}).$$

In particular every such map includes the data of a map $\overline{\phi}:[p]\to[m+1]$. To simplify notation we will denote the object $(j[p]\boxtimes\mathbf{p},\phi)$ as $B_{\phi}$ or simply $\phi$.

Let $\mathcal{C}$ denote the full subcategory of $\mathcal{D}$ consisting of the union of the following three types of objects:

1. (a)

   those $B_{\phi}$ in which $\overline{\phi}$ factors as

   $$\overline{\phi}:[p]\to\{0,\dots,m\}\subseteq[m+1],$$

2. (b)

   those $B_{\phi}$ in which $\overline{\phi}$ factors as

   $$\overline{\phi}:[p]\to\{m,m+1\}\subseteq[m+1],\quad\text{and}$$

3. (c)

   those $B_{\phi}$ in which $(\overline{\phi})^{-1}(\{m\})=\{r\}\subseteq[p]$ consists of a singleton for some $0\leq r\leq p$.

For any object $D\in\mathcal{D}$, the under category $\mathcal{C}_{D/}$ actually has an initial object and is thus weakly contractible. Consequently (see, e.g., [28, Th. 4.1.3.1 and Proposition 4.1.1.8]), the induced morphism of (homotopy) colimits over these categories is an equivalence; in particular, it follows that the following canonical maps are equivalences in $\pre(\Delta^{\times n})$:

$$\colim_{\mathcal{C}}B\simeq\colim_{\mathcal{D}}B\simeq\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m}).$$

For each $\phi\in\mathcal{C}$, let $A_{\phi}$ denote the fiber product

$$A_{\phi}:=B_{\phi}\times_{\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m})}\left(\delta_{n}^{*}\delta_{n}(j[m]\boxtimes\mathbf{m})\cup^{j[0]\boxtimes\mathbf{0}}\delta_{n}^{*}\delta_{n}(j[1]\boxtimes\mathbf{m})\right).$$

This gives rise to a new functor $A:\mathcal{C}\to\pre(\Delta^{\times n})$, and as colimits in $\pre(\Delta^{\times n})$ are universal, we have natural equivalences:

	$\displaystyle\colim_{\mathcal{C}}A$	$\displaystyle\simeq\left(\colim_{\mathcal{C}}B\right)\times_{\delta_{n}^{*}\delta_{n}(j[m+1]\boxtimes\mathbf{m})}\left(\delta_{n}^{*}\delta_{n}(j[m]\boxtimes\mathbf{m})\cup^{j[0]\boxtimes\mathbf{0}}\delta_{n}^{*}\delta_{n}(j[1]\boxtimes\mathbf{m})\right)$
		$\displaystyle\simeq\delta_{n}^{*}\delta_{n}(j[m]\boxtimes\mathbf{m})\cup^{j[0]\boxtimes\mathbf{0}}\delta_{n}^{*}\delta_{n}(j[1]\boxtimes\mathbf{m}).$

Thus the desired result follows if we can demonstrate that the natural map

$$\colim_{\mathcal{C}}A\to\colim_{\mathcal{C}}B$$

is in the class $T_{\Delta^{\times n}}$. This class, being saturated, is closed under colimits, and so it suffices to show that each of the maps

$$A_{\phi}\to B_{\phi}$$

is in $T_{\Delta^{\times n}}$. If $B_{\phi}\in\mathcal{C}$ is of type (a) or type (b), then $A_{\phi}\simeq B_{\phi}$ is an equivalence, hence in the desired class. If $B_{\phi}=(j[p]\boxtimes\mathbf{p},\phi)$ is of type (c), so that $(\overline{\phi})^{-1}(\{m\})=\{r\}$ for $0\leq r\leq p$, then a direct calculation reveals:

$$A_{\phi}\simeq\left(j\{0,\dots,r\}\boxtimes\mathbf{p}\right)\cup^{\left(j\{r\}\boxtimes\mathbf{p}\right)}\left(j\{r,r+1,\dots,p\}\boxtimes\mathbf{p}\right)\to j[p]\boxtimes\mathbf{p}\simeq B_{\phi}.$$

As this is one of the generators of $T_{\Delta^{\times n}}$, the result follows. ∎