Definition 6.9.1. Let $X$ be a topological space. Let $\mathcal{C}$ be a category with products. A presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $X$ is a *sheaf* if for every open covering the diagram

\[ \xymatrix{ \mathcal{F}(U) \ar[r] & \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \nolimits _{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) } \]

is an equalizer diagram in the category $\mathcal{C}$.

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