I haven’t fallen off the face of the earth – but I have been extremely busy learning lots of new maths, and I’ve had very little time to write down any of it on here. I wanted to write a short(ish) post on a really nice new topic I’ve been learning about – Galois categories. Here are two examples of where the theory comes from:

In Galois theory, we consider a field $K$ and the field extensions $L/K$. We can look at automorphisms of these extensions (automorphisms of $L$ fixing $K$ pointwise), and develop the notion of a Galois group of the extension from here. There are nice relationships between subextensions of a field extension and subgroups of the Galois group of that extension. In particular, the Galois groups of “Galois subextensions” $L/M/K$ are quotient groups of $\text{Gal}(L/K)$. One can build a category out of the extensions of $K$ (actually generalised extensions, allowing $K$-algebras that are not fields) and consider the (contravariant) functor

$\text{Hom}(-, K_s)$

on this category, where $K_s$ is a choice of separable closure of $K$. Then for things called finite étale algebras ($K$-algebras isomorphic to a finite direct product of separable extensions of $K$), this functor induces a categorical equivalence between the (opposite) category of finite étale algebras and the category of finite sets equipped with a continous left action by the Galois group $G =\text{Gal}(K_s/K)$. Separable extensions correspond to sets with a transitive $G$-action and Galois extensions correspond to $G$-sets isomorphic to finite quotient groups of $G$.

In topology, there is a similar story with covering spaces of a fixed space $X$. These are spaces that map onto $X$ via “local homeomorphisms”. The fundamental group $\pi_1 (X,x)$ acts on elements of the fibre of the covering $p: Y\to X$ above $x$ via ”lifting” (mondromy) of loops to paths. If this action is transitive, the covering is called Galois. For sufficiently nice spaces $X$, there is a simply connected covering space $\tilde{X}$ with a distinguished point $\tilde{x}$ whose group of deck transformations (automorphisms of it as a cover of $X$) is isomorphic to the fundamental group $\pi_1 (X,x)$. This is called the universal cover of $X$, and in fact it actually represents the functor

$F$

on the category of covers that sends each covering $p: Y\to X$ to its fibre above $x$. For obvious reasons this functor is called the fibre functor. Then the fibre functor induces an equivalence between the category of finite covers of $X$ and the category of finite sets equipped with a continuous left $\widehat{\pi_1 (X,x)}$-action, where $\widehat{G}$ denotes the profinite completion.

There are other examples of similar instances across mathematics; it would seem reasonable to think that the cause of this behaviour – a category with a certain type of structure equipped with a “fibre functor” to finite sets – is essentially categorical, having nothing to do really with the objects involved, be they field extensions or covers of a space.

Galois categories are the answer: both the examples above (with the opposite category taken for field extensions) turn out to be instances of Galois categories. A Galois category is a category $\mathcal{C}$ equipped with a functor $F$ taking values in finite sets such that

1. Finite fibre products exist in $\mathcal{C}$ and $\mathcal{C}$ has a terminal object;
2. Finite coproducts exist in $\mathcal{C}$ and categorical quotients by finite automorphism groups exist;
3. Every morphism $u: Y\to X$ factors as $Y\to X'\to X$, where the first arrow is a strict epimorphism (think “covering”) and the second arrow is a monomorphism that is an isomorphism onto a direct summand of $X$ (so the whole morphism is a “covering of a connected component” of $X$);
4. The functor $F$ commutes with fibre products and sends the terminal object to a singleton;
5. $F$ commutes with finite coproducts, quotients by automorphisms and sends strict epimorphisms to strict epimorphisms;
6. $F$ reflects isomorphisms.

This list of axioms is actually amazingly strong. From it we can deduce many things:

1. The decomposition of morphisms in axiom 3. is essentially unique (unique up to a unique isom);
2. $\mathcal{C}$ is “Artinian” in the sense that chains of monomorphisms $\dots \to X_2\to X_1\to X_0$ eventually become isomorphisms;
3. Every object has an essentially unique decomposition into a finite number of “connected components” (objects that are not isomorphic to a coproduct with neither summand the initial object);

Lots of these follow from exploiting the relation between $\mathcal{C}$ and finite sets via the fibre functor. Then morphisms to/from connected objects are heavily constrained:

1. Morphisms to connected objects are strict epimorphisms (since a connected object has one connected component, you must cover the whole component);
2. Strict epimorphisms from a connected object force the codomain to also be connected (i.e. if you cover an object by a connected object then the target must also be connected);
3. All endomorphisms of a connected object are automorphisms.

The last, in particular, is very reminiscent of classical Galois theory – any endomorphism of an algebraic extension of a field $K$ is an automorphism.

There is also a categorical notion of a Galois object – this is a connected object $X$ whose quotient by its automorphisms is a terminal object (equivalently there is a bijection between the automorphisms of $X$ and the elements in $F(X)$). Every connected object has a Galois closure, just as in Galois theory.

Let $X_0$ be a Galois object. By restricting to the full subcategory $\mathcal{C}^{X_0}$ of $\mathcal{C}$ consisting of objects all of whose connected components are “dominated” by $X_0$ (i.e. there is a morphism from $X_0$ to all of the connected components) and restricting the fibre functor accordingly, we obtain an isomorphism of functors

$\text{Hom}_{\mathcal{C}^{X_0}} (X_0, -)\xrightarrow{\sim} F\vert_{\mathcal{C}^{X_0}}$

and furthermore $F\vert_{\mathcal{C}^{X_0}}$ factors through an equivalence between $\mathcal{C}^{X_0}$ and the category of finite sets equipped with a continuous left action by the opposite group of the automorphism group of $X_0$, which is also the group of automorphisms of the restricted fibre functor $F\vert_{\mathcal{C}^{X_0}}$. This is the famous “Galois correspondence” of Galois theory in this very abstract language.

The goal is then to construct a “profinite” version of this equivalence with the original category $\mathcal{C}$, allowing for many more Galois correspondences. This is done by piecing together the “finite versions” with the categories $\mathcal{C}^{X_0}$. By setting

$\pi_1(\mathcal{C};F):=\text{Aut}(F)$

to be the automorphism group of the fibre functor, the above then allows us to reconstruct the topological fundamental group as the “fundamental group” of the Galois category of covers of a space/the Galois group $\text{Gal}(K_s/K)$ as the fundamental group of the opposite category of finite étale $K$-algebras, and furthermore allows us to carry both of these ideas over to schemes via the étale fundamental group.

It’s been sketchy, because I’m still digesting the ideas, but I hope to spend more time on this soon once I have read more. If you’re interested and want a very nice introduction to these ideas (you only need basic category theory!), I’d recommend getting hold of a copy of Arithmetic and Geometry Around Galois Theory and checking out the Galois Categories article by Cadoret.