The Stacks project

Lemma 24.3.2. Let $\mathcal{C}$ be a site.

  1. A composition of coverings in $\text{SR}(\mathcal{C})$ is a covering.

  2. If $K \to L$ is a covering in $\text{SR}(\mathcal{C})$ and $L' \to L$ is a morphism, then $L' \times _ L K$ exists and $L' \times _ L K \to L'$ is a covering.

  3. If $\mathcal{C}$ has products of pairs, and $A \to B$ and $K \to L$ are coverings in $\text{SR}(\mathcal{C})$, then $A \times K \to B \times L$ is a covering.

Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then (1) and (2) holds for $\text{SR}(\mathcal{C}, X)$ and (3) holds if $\mathcal{C}$ has fibre products.

Proof. Part (1) is immediate from the axioms of a site. Part (2) follows by the construction of fibre products in $\text{SR}(\mathcal{C})$ in the proof of Lemma 24.2.3 and the requirement that the morphisms in a covering of $\mathcal{C}$ are representable. Part (3) follows by thinking of $A \times K \to B \times L$ as the composition $A \times K \to B \times K \to B \times L$ and hence a composition of basechanges of coverings. The final statement follows because $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$. $\square$


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