The Stacks project

Lemma 25.12.5. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a continuous functor. Assume $\mathcal{D}$ and $\mathcal{C}$ have fibre products and $u$ commutes with them. Let $Y \in \mathcal{D}$ and $K \in \text{SR}(\mathcal{D}, Y)$ a hypercovering of $Y$. Then $u(K)$ is a hypercovering of $u(Y)$.

Proof. This is easier than the proof of Lemma 25.12.4 because the notion of being a hypercovering of an object is stronger, see Definitions 25.3.3 and 25.3.1. Namely, $u$ sends coverings to coverings by the definition of a morphism of sites. It suffices to check $u$ commutes with the limits used in defining $(\text{cosk}_ n \text{sk}_ n K)_{n + 1}$ for $n \geq 1$. This is clear because the induced functor $\mathcal{D}/Y \to \mathcal{C}/X$ commutes with all finite limits (and source and target have all finite limits by Categories, Lemma 4.18.4). $\square$

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