ScalingStacks

Proof. By Lemma 13.7, it is enough to check that the generators of $\mathrm{Segal}_{\Theta_{n}}$ and $\mathrm{Comp}_{\Theta_{n}}$ of $T_{\Theta_{n}}$ are contained in $T_{\Theta_{n}}^{(c)}$. For that assume that $[U\to V]$ is a generator of $T_{\Theta_{n}}$ and let $C_{j}\hookrightarrow C_{i}$ be an inclusion. We wish to demonstrate that for all $V\to C_{i}$ we have that

(13.8.1)

$$U\times_{C_{i}}C_{j}\to V\times_{C_{i}}C_{j}$$

is in $T_{\Theta_{n}}$. Recall that

$$\mathrm{Comp}_{\Theta_{n}}=\iota_{!}\mathrm{Comp}_{\Delta}\cup\sigma_{!}\mathrm{Comp}_{\Theta_{n-1}}\textrm{\quad and\quad}\mathrm{Segal}_{\Theta_{n}}=\mathrm{Segal}_{\Theta_{n}}\cup\sigma_{!}\mathrm{Segal}_{\Theta_{n-1}}.$$

There are several cases:

1. (a)

   $i=j$. In this trivial case (13.8.1) reduces to $[U\to V]\in T_{\Theta_{n}}$.

2. (b)

   The morphism $V\to C_{i}$ factors as $V\to C_{0}\to C_{i}$. In this case the fiber product $C_{j}\times_{C_{i}}C_{0}$ is either $C_{0}$ or empty. In the latter case (13.8.1) is an isomorphism, and in the former case it is $[U\to V]\in T_{\Theta_{n}}$. Notice that this case covers $\iota_{!}\mathrm{Comp}_{\Delta}$.

3. (c)

   $j=0<i$ and the map $V\to C_{i}$ does not factor through $C_{0}$. In this case it follows that $[U\to V]$ is not in $\iota_{!}\mathrm{Comp}_{\Delta}$, and hence

   $$V=([m];o_{1},...,o_{m})$$

   is representable with $m\neq 0$. Moreover the map

   $$V\to C_{i}=([1];C_{i-1})$$

   consists of a surjective map $[m]\to[1]$ (which specifies a unique $0<k\leq m$; the inverse image of $0\in[1]$ consists of all elements strictly less than $k$) together with map $o_{k}\to C_{i-1}$. A direct calculation shows that in this situation (13.8.1) is either an isomorphism or a generator of $T_{\Theta_{n}}$.

4. (d)

   $0<j\leq i$, the map

   $$[U\to V]\in\sigma_{!}\mathrm{Comp}_{\Theta_{n-1}}\cup\sigma_{!}\mathrm{Segal}_{\Theta_{n-1}}$$

   is a suspension, and the map $V\to C_{i}$ does not factor through $C_{0}$. It follows that $V\to C_{i}$ is the suspension of a map. This case then follows by induction and Lemma 13.3.

5. (e)

   The final case is when $0<j<i$, the map $[U\to V]\in Se_{\Theta_{n}}$ is not a suspension, nor in $\mathrm{Comp}_{\Theta_{n}}$, and the map $V\to C_{i}$ does not factor through $C_{0}$. In this case (13.8.1) is a map of the form described in Lemma 13.5. ∎