Lemma 60.18.4. Let $\mathcal{C}$ be a category endowed with the chaotic topology. Let $X$ be an object of $\mathcal{C}$ such that every object of $\mathcal{C}$ has a morphism towards $X$. Assume that $\mathcal{C}$ has products of pairs. Then for every abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ the total cohomology $R\Gamma (\mathcal{C}, \mathcal{F})$ is represented by the complex

\[ \mathcal{F}(X) \to \mathcal{F}(X \times X) \to \mathcal{F}(X \times X \times X) \to \ldots \]

associated to the cosimplicial abelian group $[n] \mapsto \mathcal{F}(X^ n)$.

**Proof.**
Note that $H^ q(X^ p, \mathcal{F}) = 0$ for all $q > 0$ as any presheaf is a sheaf on $\mathcal{C}$. The assumption on $X$ is that $h_ X \to *$ is surjective. Using that $H^ q(X, \mathcal{F}) = H^ q(h_ X, \mathcal{F})$ and $H^ q(\mathcal{C}, \mathcal{F}) = H^ q(*, \mathcal{F})$ we see that our statement is a special case of Cohomology on Sites, Lemma 21.13.2.
$\square$

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