## 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.

Lemma 18.2.1. Let $\mathcal{C}$ be a category.

1. All limits and colimits exist in $\textit{PAb}(\mathcal{C})$.

2. All limits and colimits commute with taking sections over objects of $\mathcal{C}$.

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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