Lemma 13.10 (0MKJ) · ScalingStacks

Lemma 13.10. For each $x\in\{a,b\}$ , we have

T_{\Theta_{n}}^{(x)}=\left\{[f\colon U\to V]\in T_{\Theta_{n}}\;\middle|\;\begin{aligned} &\textrm{for any }[H\to C_{i}]\in J_{x}\textrm{ and any nondegenerate }\\ &[V\to C_{i}]\in\pre(\Theta_{n})\textrm{, one has }f\times_{C_{i}}\nu H\in T_{\Theta_{n}}\end{aligned}\right\}

In other words, to verify that $f\colon U\to V$ is in one of these classes, it suffices to consider only those fiber products $f\times_{C_{i}}\nu H$ with $V\to C_{i}$ nondegenerate.

[0MKK]

Proof. Let us focus on the case $x=a$ . Let $f\colon U\to V$ be in class given on the right-hand side of the asserted identity. We wish to show that $f\in T_{\Theta_{n}}^{(a)}$ , that is for any pair of morphism $H\to C_{i}$ and $V\to C_{i}$ we have

U^{\prime}=U\times_{C_{i}}H\to V\times_{C_{i}}H=V^{\prime}

is in $T_{\Theta_{n}}$ . This follows as there exists a factorization $V\to C_{k}\to C_{i}$ and a diagram of pullbacks:

such that $V\to C_{k}$ is nondegenerate. The analogous result for $T_{\Theta_{n}}^{(b)}$ follows by the same argument and the observation that $H^{\prime}\to C_{k}$ is nondegenerate if $H\to C_{i}$ is such. ∎