Definition 59.18.4. Let $\mathcal{C}$ be a category. Given a presheaf of sets $\mathcal{G}$, we define the free abelian presheaf on $\mathcal{G}$, denoted $\mathbf{Z}_\mathcal {G}$, by the rule
\[ \mathbf{Z}_\mathcal {G}(U) = \mathbf{Z}[\mathcal{G}(U)] \]
for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ with restriction maps induced by the restriction maps of $\mathcal{G}$. In the special case $\mathcal{G} = h_ U$ we write simply $\mathbf{Z}_ U = \mathbf{Z}_{h_ U}$.
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