Lemma 11.4 (0MJV) · ScalingStacks

Lemma 11.4. Condition (R.1) is implied by the conjunction of the following.

(R.1-bis(a))

$i^{*}(S_{00})\subset T$ .
(R.1-bis(b))

For any morphism $U^{\prime}\to V^{\prime}$ of $T_{0}$ , and for any morphisms $V^{\prime}\to i^{*}(C_{i})$ and $H\to C_{i}$ with $H\in\Upsilon_{n}$ , the pullback

$U^{\prime}\times_{i^{*}(C_{i})}i^{*}H\to V^{\prime}\times_{i^{*}C_{i}}i^{*}H$

lies in $T$ .

Proof. First, consider the subclass $T^{\prime}\subset T$ containing those morphisms $U^{\prime}\to V^{\prime}$ of $T$ such that for any nondegenerate morphisms $V^{\prime}\to C_{k}$ and $H\to C_{k}$ , the pullback

U^{\prime}\times_{C_{k}}H\to V^{\prime}\times_{C_{k}}H

lies in $T$ . Since colimits in $\pre(\mathcal{R})$ are universal, one deduces immediately that the class $T^{\prime}$ is strongly saturated. Hence (R.1-bis(b)) implies that $T^{\prime}=T$ . Thus $T$ is closed under pullbacks along morphisms $H\to C_{k}$ and contains $i^{*}S_{00}$ (by (R.1-bis(a))), hence contains all of $i^{*}S_{0}$ . ∎