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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