Lemma 25.6.2. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. Then $\check{H}^0(K, \mathcal{F}) = H^0(\mathcal{G}, \mathcal{F})$.

Proof. This follows from the definition of $H^0(\mathcal{G}, \mathcal{F})$ and the fact that

$\xymatrix{ F(K_1) \ar@<1ex>[r] \ar@<-1ex>[r] & F(K_0) \ar[r] & \mathcal{G} }$

becomes an coequalizer diagram after sheafification. $\square$

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