Remark 85.17.6. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets on $\mathcal{C}$. If $\mathcal{C}$ has equalizers and fibre products, then we've defined the notion of a hypercovering of $\mathcal{G}$ in Hypercoverings, Definition 25.6.1. We claim that all the results in this section have a valid counterpart in this setting. To see this, define the localization $\mathcal{C}/\mathcal{G}$ of $\mathcal{C}$ at $\mathcal{G}$ exactly as in Sites, Lemma 7.30.3 (which is stated only for sheaves; the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{G})$ is equal to the localization of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ at the sheaf $\mathcal{G}^\# $). Then the reader easily shows that the site $\mathcal{C}/\mathcal{G}$ has fibre products and equalizers and that a hypercovering of $\mathcal{G}$ in $\mathcal{C}$ is the same thing as a hypercovering for the site $\mathcal{C}/\mathcal{G}$. Hence replacing the site $\mathcal{C}$ by $\mathcal{C}/\mathcal{G}$ in the lemmas on hypercoverings above we obtain proofs of the corresponding results for hypercoverings of $\mathcal{G}$. Example: for a hypercovering $K$ of $\mathcal{G}$ we have
for $E \in D^+(\mathcal{C}/\mathcal{G})$ where $a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{G})$ is the canonical augmentation. This is Lemma 85.17.4. Let $R\Gamma (\mathcal{G}, -) : D(\mathcal{C}) \to D(\textit{Ab})$ be defined as the derived functor of the functor $H^0(\mathcal{G}, -) = H^0(\mathcal{G}^\# , -)$ discussed in Hypercoverings, Section 25.6 and Cohomology on Sites, Section 21.13. We have
by the analogue of Cohomology on Sites, Lemma 21.7.1 for the localization fuctor $j : \mathcal{C}/\mathcal{G} \to \mathcal{C}$. Putting everything together we obtain
for $E \in D^+(\mathcal{C})$ where $g : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is the composition of $a$ and $j$.
Comments (0)