Definition 25.4.1. Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\textit{PSh}(\mathcal{C})$. By the above we get a simplicial object $\mathbf{Z}_ K^\# $ of $\textit{Ab}(\mathcal{C})$. We can take its associated complex of abelian presheaves $s(\mathbf{Z}_ K^\# )$, see Simplicial, Section 14.23. The homology of $K$ is the homology of the complex of abelian sheaves $s(\mathbf{Z}_ K^\# )$.
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