Galois category

Definition

Let $\mathcal{C}$ be a category and let $F \colon \mathcal{C} \to \mathbf{FinSet}$ be a functor to the category of finite sets. Let $q \colon Y \to X$ be a morphism of $\mathcal{C}$ such that the pullback $Y \times_X Y$ exists. The morphism $q$ is a strict epimorphism when it is the coequalizer of the two projections $Y \times_X Y \to Y$, its kernel pair. The pair $(\mathcal{C}, F)$ is a Galois category, and $F$ is a fibre functor, when these axioms hold:

1. $\mathcal{C}$ has a terminal object and all fibre products.
2. $\mathcal{C}$ has finite coproducts, in particular an initial object, and for each object $X$ and each finite subgroup $G$ of $\Aut(X)$ the quotient $X/G$ exists.
3. Each morphism $u \colon Y \to X$ factors as $u = u' \circ u''$ with $u''$ a strict epimorphism and $u'$ a monomorphism. Moreover, each monomorphism $Y \to X$ is an isomorphism of $Y$ onto a direct summand of $X$, that is, onto a coproduct component.
4. $F$ sends a terminal object to a terminal object and preserves fibre products.
5. $F$ preserves finite coproducts, sends strict epimorphisms to surjections, and preserves quotients by finite groups of automorphisms.
6. If $u$ is a morphism of $\mathcal{C}$ and $F(u)$ is a bijection, then $u$ is an isomorphism.