In this post I want to examine how several situations with nice symmetry groups (for example, Galois groups or fundamental groups) can be explained by the abstract study of things called fibre functors. My main motivating example will be the étale fundamental group of a scheme, but this somehow makes it natural to first consider classical fundamental groups of topological spaces and Galois groups of field extensions, both of which can also be explained abstractly, in a way that is fairly interesting. I suppose the most surprising aspect of this theory to me is that it seems to supply the correct definition in all three cases, in spite of the three situations (or at least the classical vs etale fundamental groups) being rather different.

Firstly, let us consider the theory of fields. Indeed, let us fix a field ${k}$, and consider the category ${\mathcal{C}}$ of its finite separable extensions. Given any separable closure ${k^s}$, we can define a fairly natural functor ${\mathcal{C}^{op} \rightarrow (Sets)}$ by

$\displaystyle Fib_{k_s}: F \mapsto Hom_k(F,k^s).$

Having defined this functor, is it natural to ask how much of a choice we were making by choosing a separable closure? In other words, what possible symmetries were there in the original situation? More formally, we can investigate the group of automorphisms of this functor. What exactly does this mean? Well, I guess what we are asking for is some kind of compatible collection of automorphisms on each ${Hom_k(F,k^s)}$. Anyone who knows classical Galois theory will be able to see that ${Gal(k^s/k)}$ is certainly contained in this group, and in fact it turns out that this is the whole group!

How can we see this? An important feature of this theory is that there are not-quite-canonical injections ${Aut_k(F) \hookrightarrow Hom_k(F,k^s)}$ for each ${F \in \mathcal{C}}$. Indeed, this is a reasonably obvious consequence of the fact that ${F \rightarrow k^s}$ is always injective, but it is a crucial property which we will call the rigidity property. Another key feature is the existence of Galois closures: for any ${F}$, there is some ${L}$ with a morphism ${F \rightarrow L}$ and with the property that ${Aut_k(L) \rightarrow Hom_k(L,k^s)}$ is in fact a bijection. It follows from this that actually ${Hom_k(F,L) \cong Hom_k(F,k^s)}$! Indeed, restricting domains gives natural surjections, and one easily checks they give an induced bijection.

Given these facts, it is possible to build’ ${Fib_{k_s}}$ out of explicit elements of ${\mathcal{C}}$, namely the Galois elements. Why are they so important? The fact that ${Aut_k(L) \rightarrow Hom_k(L,k^s)}$ is surjective implies that once we have embedded ${L}$ into ${k^s}$, any other embedding is merely an automorphism of our original embedding! In particular, the actual field ${L}$ sits naturally inside ${k^s}$. This is enough to ensure that we can fix an arbitrary system of embeddings ${\phi_L: L\hookrightarrow k^s}$, which can be made into a directed system (in our case just a nested system of subfields of ${k^s}$) by appropriate fixed choices of ${\phi_{LL'}: L \hookrightarrow L'}$.

Once we have constructed this system, it is fairly straightforward to check that there is a functorial isomorphism

$\displaystyle Fib_{k_s}(F) \cong \text{lim}_{\rightarrow L} Hom_k(F,L).$

Indeed, by the existence of Galois closures, we can see that for any given ${F}$ an element of the LHS is in fact an element of ${Hom_k(F,L)}$ composed with our fixed map ${\phi_L: L \rightarrow k^s}$, and conversely by the compatibilities chosen, an element of the RHS gives a well-defined element of the LHS.

From this description, it is then one further check that actually the automorphism group of the functor is ${lim_\leftarrow Gal(L/k) =: Gal(k^s/k)}$ and so we have identified our classical automorphism group with the automorphism group of ${Fib_{k^s}}$. In other words, studying the fibre functor at ${k^s}$ gaves us an important group associated to the field ${k}$ with some kind of weak dependence also on the choice ${k^s}$ of separable closure.

What does the moral seem to be: if you have a fairly natural looking functor from your category to the category of sets, the group of automorphisms of this functor can be extremely interesting.

Let us move on to algebraic topology. Consider a nice topological space ${X}$, and consider the category ${Cov(X)}$ of covering spaces of ${X}$. There is if anything a somewhat more obvious functor here (and one which explains the name fibre functor’ better). Given a point ${x \in X}$, we can define

$\displaystyle Fib_x: (f: Y \rightarrow X) \mapsto f^{-1}(x).$

Again, there’s a rigidity property (obtained by proving path lifting lemmas, etc.): any morphism ${Y'\rightarrow Y}$ which fixes ${X}$ is determined by its action on a single element of ${f'^{-1}(x)}$, so there is again a not-quite canonical injection ${Aut(Y/X) \rightarrow Fib_x(Y)}$. What are the automorphisms of this functor? Well, in fact in this case, using the rigidity property and the construction of the universal cover, the functor is actually representable by the universal cover ${X_x \rightarrow X}$:

$\displaystyle Fib_x(Y) \cong Hom_X(X_x,Y).$

Therefore (if you like) by the fully-faithfulness of the Yoneda embedding the automorphisms of the fibre functor are precisely ${Aut(X_x/X)}$. But we know from classical topology that this is the fundamental group ${\pi_1(X,x)}$ of the space, so once again we have constructed an important invariant of ${X}$ associated loosely to the point ${x}$.

Our final example is in some sense a combination of the previous two examples, in that it actually generalises the first but superficially resembles the second. Most importantly however, it illustrates how by analysing the above examples as we did, we can define this group relatively naturally.

Let ${S}$ be a fixed scheme, and we in some sense want to study covering spaces of ${S}$. However, examples like the punctured complex plane (and the fact its universal cover fails to be defined algebraically) convince us that because algebra is naturally somewhat more finite’ than topology, we should restrict to finite covers, and by analogy with the theory of Riemann spaces we want our covers to be smooth everywhere (there should be no points of ramification or other oddness). So it turns out a vaguely plausible thing to look at (because it fits our description just given) is the category of finite étale maps ${X \rightarrow S}$, denoted ${FEt_S}$.

We want to define a fibre functor. One of the properties our fibres had in previous examples is that the objects used to define them had to be really boring – in particular have no interesting covers themselves. Indeed, in the first example we used an algebraically closed field, which doesn’t admit any finite extensions, and the second example we unwittingly used was a single point, which is famously simply connected.

What kind of schemes could we use to define fibre functors on ${FEt_S}$? The obvious choice, in analogy with the fundamental group theory, would be to take the residue fields of closed points ${Spec k(s) \rightarrow S}$, and consider their lifts to ${X}$. However, these fields need not be algebraically closed, and so would possess some internal nontrivial symmetry behaviour which would render the theory potentially quite complicated (though I haven’t tried, maybe this could be fun!). Scared of accidentally getting big absolute Galois groups as error terms’, we instead define geometric points, maps ${s: Spec k \rightarrow S}$ with ${k}$ algebraically closed. Then, having fixed a geometric point (and it’s common to write ${s}$ to also denote ${Spec k}$ understood to be an ${S}$-scheme via ${s}$), we define

$\displaystyle Fib_s: (X \rightarrow S) \mapsto Hom_S(s,X).$

And once again, you guessed it, we have a rigidity property, this time as a result of technical properties of étale morphisms (and that finite morphisms are in particular separated), and we have the existence of Galois closures. In this case, as in the field theory case, the functor is not representable (unsurprisingly given the finiteness constraint), but it can be written as a direct limit of representable functors. In fact, the situation generalises the field theoretic situation and the derivation of the theory is almost identical (with all arrows reversed of course!), with the automorphism group of the fibre functor being what we define as the étale fundamental group ${\pi_1(S, s)}$. The argument we gave in the field theory case implies that this group is the inverse limit over the opposite automorphism groups of all finite Galois covers.

I find this abstract way to set things up using fibre functors quite surprising, and wonder which other objects out there could be invented using this method. It would be nice to understand a bit more whether we can deduce anything purely formally about fibre functors using the existence of Galois closures and rigidity, though the lack of similarity between classical and étale fundamental groups makes me unsure what kind of properties we would expect.

In all the situations outlined above, there is also some kind of Galois correspondence, and it would be good to know the extent to which this is connected with the fibre functor formalism. Certainly in the equivalent linearised situation one has Tannaka-Krein duality which in some sense gives a Galois correspondence formally from studying the fibre functor, but it seems that in all the proofs I have seen for these examples Galois correspondences are derived by more direct means (explicitly constructing objects which need to exist using descent theory or whatever).

Finally an intriguing feature is how often one must make arbitrary choices and how they always seem to be slightly but not very significant. Picking a fibre functor involves picking a basepoint, and in the end the results you get tend to be noncanonically isomorphic. Also, our injections from automorphism groups into fibres are maps from a group into a space, so aren’t canonically defined unless we pick a distinguished point in each fibre, and when we prove representability of fibre functors we always have to pick distinguished points in the representing object. I don’t understand how this laxness in the system really affects the mathematics going on, but it feels like on one level it’s crucial in that these vaguenesses are precisely the symmetries we’re measuring. As always, any thoughts would be appreciated.