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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $F \\colon \\mathcal{C} \\to \\mathbf{FinSet}$ be a [[functor|28NQ0KRHD3]] to the category of finite sets.\nLet $q \\colon Y \\to X$ be a morphism of $\\mathcal{C}$ such that the [[pullback|Q05K04QJ71]] $Y \\times_X Y$ exists.\nThe morphism $q$ is a *strict epimorphism* when it is the [[coequalizer|DVSTWGVCSZ]] of the two projections $Y \\times_X Y \\to Y$, its [[kernel pair|E70DQGM8X5]].\nThe pair $(\\mathcal{C}, F)$ is a *Galois category*, and $F$ is a *fibre functor*, when these axioms hold:\n\n1. $\\mathcal{C}$ has a [[terminal object|EK85VSSRMR]] and all fibre products.\n2. $\\mathcal{C}$ has finite [[coproducts|4FJ0S71P58]], in particular an initial object, and for each object $X$ and each finite subgroup $G$ of $\\Aut(X)$ the quotient $X/G$ exists.\n3. Each morphism $u \\colon Y \\to X$ factors as $u = u' \\circ u''$ with $u''$ a strict epimorphism and $u'$ a [[monomorphism|Y2NP1JFR86]].\nMoreover, each monomorphism $Y \\to X$ is an isomorphism of $Y$ onto a direct summand of $X$, that is, onto a coproduct component.\n4. $F$ sends a terminal object to a terminal object and preserves fibre products.\n5. $F$ preserves finite coproducts, sends strict epimorphisms to surjections, and preserves quotients by finite groups of automorphisms.\n6. If $u$ is a morphism of $\\mathcal{C}$ and $F(u)$ is a bijection, then $u$ is an isomorphism.\n\n## Remarks\n\nHere $X/G$ means the coequalizer of the family of maps $g \\colon X \\to X$, $g \\in G$.\nExample: for a [[profinite group|KA0YDB6QXE]] $\\pi$, the category of finite sets with a continuous action of $\\pi$, with $F$ the functor that forgets the action, satisfies all six axioms.\nMain theorem: for every Galois category $(\\mathcal{C}, F)$, the group $\\pi = \\Aut(F)$ is profinite and $F$ induces an equivalence of $\\mathcal{C}$ with finite continuous $\\pi$-sets.\nNon-example: the category of all finite sets with $F$ the functor that doubles each set, $F(X) = X \\sqcup X$, fails axiom 4, since it sends the one-point set to a two-point set.\nThe axioms follow Grothendieck's SGA 1; for a field $k$, finite étale $k$-algebras with arrows reversed form a Galois category whose group is the absolute Galois group of $k$.\n",
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