The Stacks project

Lemma 6.4.3. Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of sets. Consider the following types of structure on $\mathcal{F}$:

1. For every open $U$ the structure of an abelian group on $\mathcal{F}(U)$ such that all restriction maps are abelian group homomorphisms.

2. A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$, a map of presheaves $- : \mathcal{F} \to \mathcal{F}$ and a map $0 : * \to \mathcal{F}$ (see Example 6.4.1) satisfying all the axioms of $+, -, 0$ in a usual abelian group.

3. A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$, a map of presheaves $- : \mathcal{F} \to \mathcal{F}$ and a map $0 : * \to \mathcal{F}$ such that for each open $U \subset X$ the quadruple $(\mathcal{F}(U), +, -, 0)$ is an abelian group,

4. A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ such that for every open $U \subset X$ the map $+ : \mathcal{F}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an abelian group.

There are natural bijections between the collections of types of data (1) - (4) above.