Definition 59.11.1. A presheaf $\mathcal{F}$ of sets (resp. abelian presheaf) on a site $\mathcal{C}$ is said to be a separated presheaf if for all coverings $\{ \varphi _ i : U_ i \to U\} _{i\in I} \in \text{Cov} (\mathcal{C})$ the map

$\mathcal{F}(U) \longrightarrow \prod \nolimits _{i\in I} \mathcal{F}(U_ i)$

is injective. Here the map is $s \mapsto (s|_{U_ i})_{i\in I}$. The presheaf $\mathcal{F}$ is a sheaf if for all coverings $\{ \varphi _ i : U_ i \to U\} _{i\in I} \in \text{Cov} (\mathcal{C})$, the diagram

59.11.1.1
\begin{equation} \label{etale-cohomology-equation-sheaf-axiom} \xymatrix{ \mathcal{F}(U) \ar[r] & \prod _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{i, j \in I} \mathcal{F}(U_ i \times _ U U_ j), } \end{equation}

where the first map is $s \mapsto (s|_{U_ i})_{i\in I}$ and the two maps on the right are $(s_ i)_{i\in I} \mapsto (s_ i |_{U_ i \times _ U U_ j})$ and $(s_ i)_{i\in I} \mapsto (s_ j |_{U_ i \times _ U U_ j})$, is an equalizer diagram in the category of sets (resp. abelian groups).

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