Definition 58.3.6. Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$ be a functor. The pair $(\mathcal{C}, F)$ is a *Galois category* if

$\mathcal{C}$ has finite limits and finite colimits,

every object of $\mathcal{C}$ is a finite (possibly empty) coproduct of connected objects,

$F(X)$ is finite for all $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and

Here we say $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is connected if it is not initial and for any monomorphism $Y \to X$ either $Y$ is initial or $Y \to X$ is an isomorphism.

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