The Stacks project

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

\[ L : U \longmapsto \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i(U) \]


\[ C : U \longmapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(U). \]

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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