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

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

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