In SGA1 Grothendieck introduced the notion of a Galois category to study/define/characterise the category of finite etale morphisms over a scheme and the associated etale fundamental group of a scheme.
All examples of Galois categories that I've heard of have come from very geometric sources: locally constant morphisms of schemes, tamely ramified morphisms etc. I suspect this is just because I only study algebraic geometry. But maybe not.
Question: Are there examples of Galois categories that arise in situations that aren't, at least at first sight, "obviously geometric"?
I have a feeling they should crop up in logic, but that's just from some smidge of understanding of topos theory and the relation it has with logic.
I say "aren't obviously geometric" because I believe that once you know there is a Galois category hanging around, then there is probably some geometric picture you could make for the objects of the category. I could be wrong about that too!
Any examples and references would be appreciated. Thanks!
Alright, I dug up my copy of Johnstone's Topos Theory and I figured I'd need to go into a little more detail than a comment could hold. My reference for this information, by the way, chiefly comes from Section 8.4 of Peter Johnstone's 1977 monograph Topos Theory (or at least the 2014 Dover reprinting at any rate). For this we'll need to go back and characterize/describe Boolean toposes, which intuitively should be those elementary toposes which have an internal logic which is classical. I'll go through below and define some topos theoretic conventions we'll need below. I will give a tl;dr however.
tl;dr Galois categories are exactly the small Boolean toposes $\mathcal{G}$ for which there is an exact isomorphism-reflecting functor $F:\mathcal{G} \to \mathbf{FinSet}$; equivalently these are the toposes equivalent to the category of continuous finite $G$-sets for a profinite group $G$.
Definition I'll write the terminal object of a category $\mathscr{C}$ as $\top$. This is because it's nice to think of $\top$ as a truth value (and the top of the category). Similarly, the initial object of a category $\mathscr{C}$ will be written as $\bot$ (it's nice to think of it as a false value and the bottom of the category).
Definition The subobject classifier of a topos $\mathcal{E}$ will be denoted $\Omega$. This is an object $\Omega$ equipped with a morphism $\operatorname{true}:\top \to \Omega$ for which given any monic $m:A \to B$ in $\mathcal{E}$ there is a unique morphism $\chi_{m}:B \to \Omega$ (the classifying map of $m$) for which the diagram $$ \begin{array} & A& \xrightarrow{!_A} & \top \\ m\downarrow & & \downarrow \operatorname{true} \\ B & \xrightarrow[\exists!\,\chi_m]{} & \Omega \end{array} $$ is a pullback in $\mathcal{E}$.
Definition The map $\operatorname{false}:\top \to \Omega$ in a topos $\mathcal{E}$ is defined to be the classifying map of the monic $\bot \to \top$ (note that this is monic because toposes are Cartesian closed and any map $X \to \bot$ is necessarily an isomorphism witnessing $X \cong \bot$ --- this is a cute exercise that I recommend trying).
The morphism $\operatorname{false}:\top \to \Omega$ is monic in any topos $\mathcal{E}$. By classifying this map, we get an endomorphism of $\Omega$ which allows us to think of negating truth values in $\mathcal{E}$.
Definition The negation endomorphism $\neg:\Omega \to \Omega$ is defined to be the classifying map of $\operatorname{false}:\top \to \Omega$.
Definition/Proposition An elementary topos $\mathcal{E}$ is said to be Boolean if and only if $\mathcal{E}$ is $\neg\neg$-stable, i.e., $\neg \circ \neg = \operatorname{id}_{\Omega}$. Equivalently, the morphisms $\operatorname{true}:\top \to \Omega$ and $\operatorname{false}:\top \to \Omega$ induce the coproduct injections in an isomorphism $\Omega \cong \top \coprod \top$.
Definition A Galois category is a pair $(\mathcal{G},F)$ where $\mathcal{G}$ is a small Boolean topos and $F:\mathcal{G} \to \mathbf{FinSet}$ is an exact isomorphism-reflecting functor.
Let me now convince you that this definition is the same (or at least allows us to recapture) the Galois categories of SGA 1. Recall that if $G$ is a topological group, $\mathbf{Cont}(G)$ is the category of continuous (left) $G$-sets (so the left $G$-sets $X$ for which given any $x \in X$, the stabilizer subgroup $\operatorname{Stab}(x)$ is an open subgroup of $G$. We'll also write $\mathbf{Cont}_{\mathbf{Fin}}(G)$ for the (sub)category of finite continuous left $G$-sets.
Proposition Let $(\mathcal{G},F)$ be a Galois category in the sense above. Then every object $A$ in $\mathcal{G}$ is uniquely expressible as a product of indecomposables (atoms) and any endomorphism of an indecomposable (atom) is an automorphism.
Sketch For the first claim, we can ignore the case that $X \cong \bot$ as in this case $$ X \cong \coprod_{x \in \emptyset} A_x $$ where each $A_x$ is indecomposable. For $X \not\cong \bot$, if $X$ is an atom we're done. If $X$ is not an atom, write $X \cong X_1 \coprod X_2$ and use $F$ to decompose $FX \cong FX_1 \coprod FX_2$. Since $FX$ is a finite set, this process terminates after finitely many steps. To conclude, use that $F$ is isomorphism reflecting to get the coproduct decomposition of $X$.
For the second claim, let $A$ be a non-intial atom and let $a:A \to A$ an endomorphism. Since $A \not\cong \bot$, $a:A \to A$ has a non-zero image (and hence is the whole of $A$, as $A$ has no non-trivial subobjects). Use $Fa:FA \to FA$ to note that $Fa$ must be a surjection from the finite set $FA$ to the finite set $FA$; thus $Fa$ is an automorphism, and again by $F$ isomorphism-reflecting we get that $a$ is an automorphism. $\square$
The next trick we use is to show that the functor $F$ is a fibre functor like how you'd expect. Namely, it looks like the automorphisms of covers over some geometric point. To do this, we get the following proposition which I won't prove here due to already making this answer too long (although I may feel up to finishing the proof later). We'll also need to establish after that there are normal objects in the category where the pro-representable colimit stabilize, which essentially says that the automorphisms of the cover are determined after some stage of how they manipulate the finite covers.
Proposition Let $(\mathcal{G},F)$ be a Galois category. The functor $F:\mathcal{G} \to \mathbf{FinSet}$ is pro-representable in the sense that there is a filtered category $I$ and an inverse system $\lbrace A_i \in \mathcal{G}_0 \; | \; i \in I^{\text{op}}_0\rbrace$ for which there is a natural isomorphism $$ F(-) \cong \lim_{\substack{\longrightarrow \\ i \in I}}\mathcal{G}(A_i,-). $$
Sketch I'll construct the category $I$ in this sketch and not much else. Define the category $I$ by saying that the objects $I_0$ of $I$ are pairs $(A,a)$ where $A$ is an atom of $\mathcal{G}$ and $a \in F(A)$ while saying that the morphisms $I_1$ of $A$ are maps $f:(A,a) \to (B,b)$ where $f:A \to B$ is a morphism in $\mathcal{G}$ for which $F(f)(a) = b$. Then $I^{\text{op}}$ can be shown to be a filtered poset, since if there are morphisms $f,g:A \to B$ for which $f(a) = g(a)$, then the equalizer $\operatorname{Eq}(f,g) \not\cong \bot$ so $\operatorname{Eq}(f,g) \cong A$. The filteredness of $I^{\text{op}}$ follows from the fact that $(A,a)$ and $(B,b)$ have maps into $(C,(a,b))$ in $I$, where $C$ is the component of $A \times B$ which, under the coproduct decomposition of $A \times B$, has $(a,b) \in FC$. Note that $I \not\cong \emptyset$ because $F(\top) \cong \lbrace \ast \rbrace \not\cong \emptyset$ so $\top \not\cong \bot$ in $\mathcal{G}$. $\square$
The idea here is that each of the $A_i$ is a finite quotient of some proposed ``fundamental group'' of the topos. However, we need to do some massaging to make sure that we can make this more precise and careful. For this we note that given any object $(A,a)$ of the filtered category $I^{\text{op}}$, there is a natural map $$ \mathcal{G}(A,A) \to F(A) $$ which is monic. We say that $A$ is normal if this map is an isomorphism; because $\operatorname{Aut}(A) = \mathcal{G}(A,A)$ acts transitively on $FA$, this definition is independent of the choice of basepoint $a$. The benefit of using normal subobjects is that they give us the following characterization and theorem.
Proposition For any object $X \in \mathcal{G}_0$, there is a normal object $(A,a) \in I_0$ for which the natural map $$ \mathcal{G}(A,X) \to FX $$ is an isomorphism.
Theorem Let $\mathcal{G}$ be a small category and let $F:\mathcal{G} \to \mathbf{FinSet}$ be a functor. Then the following are equivalent: