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Different from the definition in [Exposé V, Definition 5.1, SGA1]. Compare with [Definition 7.2.1, BS].

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

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

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

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

  4. $F$ reflects isomorphisms1 and is exact2.

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.

[1] Namely, given a morphism $f$ of $\mathcal{C}$ if $F(f)$ is an isomorphism, then $f$ is an isomorphism.
[2] This means that $F$ commutes with finite limits and colimits, see Categories, Section 4.23.

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