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}.
Comments (0)