Lemma 59.18.5. Notation and assumptions as in Definition 59.18.1. The Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ can be described explicitly as follows

\begin{eqnarray*} \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) & = & \left( \prod _{i_0 \in I} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{U_{i_0}}, \mathcal{F}) \to \prod _{i_0, i_1 \in I} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}}, \mathcal{F}) \to \ldots \right) \\ & = & \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}\left( \left( \bigoplus _{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus _{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}} \leftarrow \ldots \right), \mathcal{F}\right) \end{eqnarray*}

Proof. This follows from the formula above. See Cohomology on Sites, Lemma 21.9.3. $\square$

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