## 6.8 Abelian sheaves

Definition 6.8.1. Let $X$ be a topological space.

An *abelian sheaf on $X$* or *sheaf of abelian groups on $X$* is an abelian presheaf on $X$ such that the underlying presheaf of sets is a sheaf.

The category of sheaves of abelian groups is denoted $\textit{Ab}(X)$.

Let $X$ be a topological space. In the case of an abelian presheaf $\mathcal{F}$ the sheaf condition with regards to an open covering $U = \bigcup U_ i$ is often expressed by saying that the complex of abelian groups

\[ 0 \to \mathcal{F}(U) \to \prod \nolimits _ i \mathcal{F}(U_ i) \to \prod \nolimits _{(i_0, i_1)} \mathcal{F}(U_{i_0} \cap U_{i_1}) \]

is exact. The first map is the usual one, whereas the second maps the element $(s_ i)_{i \in I}$ to the element

\[ ( s_{i_0}|_{U_{i_0} \cap U_{i_1}} - s_{i_1}|_{U_{i_0} \cap U_{i_1}} )_{(i_0, i_1)} \in \prod \nolimits _{(i_0, i_1)} \mathcal{F}(U_{i_0} \cap U_{i_1}) \]

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