Lemma 84.31.1. Let $X \to S$ be a morphism of schemes. Suppose $Y \to (X/S)_\bullet$ is a cartesian morphism of simplicial schemes. For $y \in Y_0$ a point define

$T_ y = \{ y' \in Y_0 \mid \exists \ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\}$

as a subset of $Y_0$. Then $y \in T_ y$ and $T_ y \cap T_{y'} \not= \emptyset \Rightarrow T_ y = T_{y'}$.

Proof. Combine Lemma 84.30.1 and Groupoids, Lemma 39.3.4. $\square$

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