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^\# .
18.5 Free abelian sheaves
Here is the notion of a free abelian sheaf on a sheaf of sets.
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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