Definition 7.12.1. Let \mathcal{C} be a category. We say that a family \{ U_ i \to U\} _{i \in I} is an effective epimorphism if all the morphisms U_ i \to U are representable (see Categories, Definition 4.6.4), and for any X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) the sequence
is an equalizer diagram. We say that a family \{ U_ i \to U\} is a universal effective epimorphism if for any morphism V \to U the base change \{ U_ i \times _ U V \to V\} is an effective epimorphism.
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