Lemma 18.2.1. Let \mathcal{C} be a category.
All limits and colimits exist in \textit{PAb}(\mathcal{C}).
All limits and colimits commute with taking sections over objects of \mathcal{C}.
18.2 Abelian presheaves
Let \mathcal{C} be a category. Abelian presheaves were introduced in Sites, Sections 7.2 and 7.7 and discussed a bit more in Sites, Section 7.44. We will follow the convention of this last reference, in that we think of an abelian presheaf as a presheaf of sets endowed with addition rules on all sets of sections compatible with the restriction mappings. Recall that the category of abelian presheaves on \mathcal{C} is denoted \textit{PAb}(\mathcal{C}).
The category \textit{PAb}(\mathcal{C}) is abelian as defined in Homology, Definition 12.5.1. Given a map of presheaves \varphi : \mathcal{G}_1 \to \mathcal{G}_2 the kernel of \varphi is the abelian presheaf U \mapsto \mathop{\mathrm{Ker}}(\mathcal{G}_1(U) \to \mathcal{G}_2(U)) and the cokernel of \varphi is the presheaf U \mapsto \mathop{\mathrm{Coker}}(\mathcal{G}_1(U) \to \mathcal{G}_2(U)). Since the category of abelian groups is abelian it follows that \mathop{\mathrm{Coim}}= \mathop{\mathrm{Im}} because this holds over each U. A sequence of abelian presheaves
\mathcal{G}_1 \longrightarrow \mathcal{G}_2 \longrightarrow \mathcal{G}_3
is exact if and only if \mathcal{G}_1(U) \to \mathcal{G}_2(U) \to \mathcal{G}_3(U) is an exact sequence of abelian groups for all U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}). We leave the verifications to the reader.
Proof. Let \mathcal{I} \to \textit{PAb}(\mathcal{C}), i \mapsto \mathcal{F}_ i be a diagram. We can simply define abelian presheaves L and C by the rules
L : U \longmapsto \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i(U)
and
C : U \longmapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(U).
It is clear that there are maps of abelian presheaves L \to \mathcal{F}_ i and \mathcal{F}_ i \to C, by using the corresponding maps on groups of sections over each U. It is straightforward to check that L and C endowed with these maps are the limit and colimit of the diagram in \textit{PAb}(\mathcal{C}). This proves (1) and (2). Details omitted. \square
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