Definition 6.4.4. Let $X$ be a topological space.

1. A presheaf of abelian groups on $X$ or an abelian presheaf over $X$ is a presheaf of sets $\mathcal{F}$ such that for each open $U \subset X$ the set $\mathcal{F}(U)$ is endowed with the structure of an abelian group, and such that all restriction maps $\rho ^ U_ V$ are homomorphisms of abelian groups, see Lemma 6.4.3 above.

2. A morphism of abelian presheaves over $X$ $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets which induces a homomorphism of abelian groups $\mathcal{F}(U) \to \mathcal{G}(U)$ for every open $U \subset X$.

3. The category of presheaves of abelian groups on $X$ is denoted $\textit{PAb}(X)$.

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