## 18.5 Free abelian sheaves

Here is the notion of a free abelian sheaf on a sheaf of sets.

Definition 18.5.1. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets. The free abelian sheaf $\mathbf{Z}_\mathcal {G}^\#$ on $\mathcal{G}$ is the abelian sheaf $\mathbf{Z}_\mathcal {G}^\#$ which is the sheafification of the free abelian presheaf on $\mathcal{G}$. In the special case $\mathcal{G} = h_ X$ of a representable presheaf associated to an object $X$ of $\mathcal{C}$ we use the notation $\mathbf{Z}_ X^\#$.

This construction is clearly functorial in the presheaf $\mathcal{G}$. In fact it provides an adjoint to the forgetful functor $\textit{Ab}(\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Here is the precise statement.

Lemma 18.5.2. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$, $\mathcal{F}$ be a sheaves of sets. Let $\mathcal{A}$ be an abelian sheaf. Let $U$ be an object of $\mathcal{C}$. Then we have

\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F}) & = \mathcal{F}(U), \\ \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_\mathcal {G}^\# , \mathcal{A}) & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\ \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_ U^\# , \mathcal{A}) & = \mathcal{A}(U). \end{align*}

All of these equalities are functorial.

Proof. Omitted. $\square$

Lemma 18.5.3. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets. Then $\mathbf{Z}_\mathcal {G}^\# = (\mathbf{Z}_{\mathcal{G}^\# })^\#$.

Proof. Omitted. $\square$

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