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 and the field extensions . We can look at automorphisms of these extensions (automorphisms of fixing 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” are quotient groups of . One can build a category out of the extensions of (actually generalised extensions, allowing -algebras that are not fields) and consider the (contravariant) functor

on this category, where is a choice of **separable closure** of . Then for things called **finite étale algebras** (-algebras isomorphic to a finite direct product of separable extensions of ), 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 . Separable extensions correspond to sets with a transitive -action and Galois extensions correspond to -sets isomorphic to finite quotient groups of .

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

on the category of covers that sends each covering to its fibre above . For obvious reasons this functor is called the **fibre functor**. Then the fibre functor induces an equivalence between the category of finite covers of and the category of finite sets equipped with a continuous left -action, where 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 equipped with a functor taking values in finite sets such that

- Finite fibre products exist in and has a terminal object;
- Finite coproducts exist in and categorical quotients by finite automorphism groups exist;
- Every morphism factors as , 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 (so the whole morphism is a “covering of a connected component” of );
- The functor commutes with fibre products and sends the terminal object to a singleton;
- commutes with finite coproducts, quotients by automorphisms and sends strict epimorphisms to strict epimorphisms;
- reflects isomorphisms.

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

- The decomposition of morphisms in axiom 3. is
**essentially unique**(unique up to a unique isom); - is “
**Artinian**” in the sense that chains of monomorphisms eventually become isomorphisms; - 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 and finite sets via the fibre functor. Then morphisms to/from connected objects are heavily constrained:

- Morphisms
**to**connected objects are strict epimorphisms (since a connected object has one connected component, you must cover the whole component); - 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); **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 is an automorphism.

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

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

and furthermore factors through an equivalence between and the category of finite sets equipped with a continuous left action by the opposite group of the automorphism group of , which is also the group of automorphisms of the restricted fibre functor . 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 , allowing for many more Galois correspondences. This is done by piecing together the “finite versions” with the categories . 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 as the fundamental group of the opposite category of finite étale -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.

Hey Alex,

Nice post! Cadoret’s article is certainly very well written. By and by, that volume also contains an article of Bertin which is, in my very humble opinion, amongst the very, very best texts on algebraic stacks.

I’m also curious what your opinion about the following is. I think that many people who have learned about the étale fundamental group have learned about the formalism of Galois categories if not from Cadoret then, perhaps more likely, from Lenstra’s book. That said, I really, honestly, truly don’t think about the étale fundamental group in this language. I think about it more as just, well, acting like normal topology. Of course, one formalizes this statement ‘of acting like normal topology’ by using this language of Galois categories, but I certainly don’t think this way.

To make an analogy, I think the following is apt. When I was a wee undergrad I tried to read Weibel’s text on homological algebra. Initially, the book took me *forever*. I kept trying to internalize what precisely an abelian category was and to try and prove everything from first principles (e.g. the dreaded purely abelian category (i.e. no Freyd-Mitchel embedding theorem) proof of the snake lemma). The most momentous decision I made in that reading course was to just pretend that everything is a category of modules. To try and think about things from the point of view of a ‘general abelian category’ got me nowhere. I just pretended that everything was modules and had faith that the argument went through. To this day I honestly don’t really think of an abelian category as anything but a category of modules.

So, I’m curious, do you actually think that the formalism of Galois categories is enlightening? Or, rather, is the enlightening aspect the fact that there are a set of axioms which allow you to treat some mathematical situation as covering space theory from topology guilt-free, just as abelian categories allow you (essentially) to treat very general categories ‘as if they were categories of modules’?

Also, here’s an interesting question that occurred to me while writing this. Is there any sort of analogue of the Freyd-Mitchel embedding theorem? Namely, can one abstract classify those Galois categories which are coming from covering space theory (say of a locally simply connected locally path connected space). Moreover, if such a classification can be done, is there any sort of ‘locality statement’ that says that every Galois category locally looks like (a full subcategory of) such a category?

Specifically, in the context of abelian categories, these two would pan out as:

1) An abelian category is a category of modules if it has a ‘compact progenerator’.

2) The locality condition is the Freyd-Mitchell embedding theorem: every small abelian category is a category of modules.

I wonder if results are known in this direction?

Thanks again for the nice post!

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Hi Alex, thanks for the comment! I think you’ve asked some really interesting questions.

Firstly, about how I see Galois categories: on the one hand, while reading Cadoret’s article I’ve tried to have in my mind the idea of topological covers all the time. This helped me make sense of the axioms – for example, Axiom (3) above didn’t make much sense to me until I realised what it was trying to model. And a lot of the results I’ve seen so far in the article seem are made much easier for me to understand and remember by attaching topological context to them (and, for certain things results, also imagining my objects as field extesions in the opposite category).

On the other hand, there are two reasons why I really like this abstract framework. The first is that I have quite a fondness for pure category theory and so some of the proofs, once they are entirely stripped back to the categorical mechanisms, seem to be more enlightening – or at least easier to follow – to me than the same things in Galois theory/topology. Perhaps this is just because the process of doing category theory is (at least for me) necessarily more precise than other areas of maths, where details are often glossed over. Maybe this is a sign that I’m still finding my feet in other areas of maths, but it’s one reason I like the Galois category idea anyway.

The other reason I like Galois categories is seeing how far we can get with some very basic formal axioms, so I guess this is kinda like what you said. The decomposition into a finite number of connected components, Artinian property and the very “rigid” nature of connected objects are all good examples of this. So I guess this is somehow dual to my other reason because we’re really seeing the purely formal stuff “come alive” and turn into familiar objects. Another thing I’m learning in parallel is the idea of Tannakian categories (which I am also really enjoying and will probably post about on here), and again it is great in this situation to see how far you can push abstract ideas until they turn into concrete familiar concepts like dimension of vector spaces, trace etc.

Anyway, it sounds like you had a similar experience to me with homological algebra. I recently also decided accept the embedding theorem and now that I feel like I’ve internalised it, learning the subject is a LOT easier! There are probably lots of other examples in maths where one’s conceptual view of the objects is actually a much more concrete version that what they might be, but the embedding theorem is the only theorem I know of that shows we don’t really need to worry about the distinction.

Lastly, I think your classification question about Galois categories being categories of covers is really interesting. I suppose one thing that categories of covers of a (nice) space have, that general Galois categories might not, is an honest representing object for the fibre functor – namely, the universal cover (which would be initial amongst covers if people didn’t allow coverings from the empty topological space, which is apparently accepted). This doesn’t completely characterise these categories though, because (if I’ve got this right) e.g. in the (opposite) category of finite étale -algebras, the separable closure of is and this is a real-life representing object for the fibre functor on this category. So I will have a think about what other properties categories of covers have…

Anyway, thanks for posing this very interesting (and possibly very difficult) question! I will definitely think more about it as I understand more about Galois categories. If you hear of any results on this – or have some ideas – I’d love to hear them 🙂

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