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


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


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.