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.

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