The Stacks project

Definition 18.4.1. Let $\mathcal{C}$ be a category. Let $\mathcal{G}$ be a presheaf of sets. The free abelian presheaf $\mathbf{Z}_\mathcal {G}$ on $\mathcal{G}$ is the abelian presheaf defined by the rule

\[ U \longmapsto \mathbf{Z}[\mathcal{G}(U)]. \]

In the special case $\mathcal{G} = h_ X$ of a representable presheaf associated to an object $X$ of $\mathcal{C}$ we use the notation $\mathbf{Z}_ X = \mathbf{Z}_{h_ X}$. In other words

\[ \mathbf{Z}_ X(U) = \mathbf{Z}[\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, X)]. \]

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