An extension of the Galois theory of Grothendieck

Question

This question is about Joyal and Tierney's famous An extension of the Galois theory of Grothendieck. One of the main results states (see the MathSciNet review by Peter Johnstone):

Joyal and Tierney's theorem. Each Grothendieck topos is equivalent to the topos of equivariant sheaves on a groupoid in the category of locales.

Question 1. What does this statement have to do with Grothendieck's Galois theory?

I have to admit, I don't know much about Grothendieck's Galois theory and which kind of theorems it includes. In my mind, I usually identify the term "Grothendieck's Galois theory" with the following theorem (see Chua - The étale fundamental group for a concise introduction and SGA 1 for a complete treatment):

Grothendieck's theorem. Let $X$ be a scheme and $x\in X$. Then the category of finite étale covers of $X$ is equivalent to the category of finite continuous $\pi_1(X,x)$-sets, where $\pi_1(X,x)$ is the étale fundamental group of $X$ at the base point $x$.

This is related to Galois theory because whenever $k$ is a field, then the étale fundamental group of $\operatorname{Spec}(k)$ is the absolute Galois group of $k$.

A more precise version of Question 1 would be: does Joyal and Tierney's theorem imply Grothendieck's theorem? (But an answer showing that Joyal and Tierney's theorem implies a modified version of Grothendieck's theorem would satisfy me as well.)

Question 2. In an answer by Zhen Lin to "What does a proof in an internal logic actually look like?" it is mentioned that the proof of Joyal and Tierney's theorem uses the internal language of toposes. In which way does it use the internal language? Which kind of theorems are proved internally, how are they interpreted externally, and why are the external interpretations of these theorems helpful in proving Joyal and Tierney's theorem? Skimming the text I don't see any logic at all.

Answer 1

The point of view where this title comes from is that Grothendieck's theorem can be seen as a characterization of toposes of the form $BG$ for $G$ a profinite group. It shows that some toposes can be represented as $BG$.

I think before Joyal–Tierney's paper it was also known how to generalize from profinite group to general localic groups.

Joyal–Tierney's theorem shows that if you replace "pro-finite group" by "localic groupoid" then you actually get all Grothendieck toposes this way.

You can't directly recover Grothendieck's theorem from Joyal–Tierney's theorem in the sense that the theorem as stated above doesn't tell you for which toposes the localic groupoid can be chosen to be a profinite group. But if you are familiar with the method used in the paper and how the groupoid is obtained (which in my opinion are even more important than the theorem itself) then it is fairly easy to recover Grothendieck's theorem. For example, it immediately follows from Joyal–Tierney's paper that a topos is of the form $BG$ for $G$ a localic groups if and only if it admits a point $* \to \mathcal{T}$ which is an open surjection (which does feel similar to Grothendieck's theorem in terms of a fiber functor).

Regarding the use of internal logic, it is definitely not essential, it just makes everything simpler (at least if you are ok with its use) but one could do without it.

The main way they use internal logic is that in the first sections they prove some results about sup-lattice and frames locales, that are later applied not to sup-lattices and frames, but to sup-lattices and frames in a topos $\mathcal{T}$. I believe there are also a few places where they make a claim about a morphism of locales $f:X \to Y$ and then only prove it when $Y$ is the point (sorry I don't have the paper with me to give precise reference).

Another place where one can consider they use a bit of internal logic — though this one might be only at the level of intuition — is when they show that every topos admits an open cover by a locale. They do this by considering the topos as a classifying topos of some theory $T$ and considering the propositional theory $T'$ of "enumerated $T$-models", that is, $T$-models that are explicitly given as a subquotient of the natural numbers. Though if I remember correctly, they present the argument in a way that doesn't directly involve any logic… (and in any case there are other proofs of this results that are purely in terms of sites, for example the one in MacLane and Moerdijk's book "Sheaves in geometry and logic").

Answer 2

$\DeclareMathOperator\Sh{Sh}$This is not an answer, but is too long for a comment — it is about question 1 specifically.

The point is that you'll want to consider the finite étale topos of $X$, namely you take as a site the finite étale maps $Y\to X$, and étale covers as covers; and take sheaves on this site : $\Sh(X^\text{f,ét})$. This is a Grothendieck topos, and by Joyal–Tierney's result, it is sheaves on a localic groupoid $\mathcal G_X$. Now the idea is that $\mathcal G_X$ is supposed to be $B\pi_1^\text{ét}(X,x)$ if $X$ is connected and $x$ is a point of $X$.

Indeed, suppose you know Grothendieck's theorem. Then the site I described is the site $\pi_1^\text{ét}(X,x)\text-\mathrm{FinSet}$ with topology given by the effective epimorphism topology, i.e. surjections of underlying (finite) sets.

Now in general, if $G$ is a profinite group, and $BG$ the associated localic groupoid, then sheaves on $BG$ are the same as sheaves on $G\text-\mathrm{FinSet}$ with the effective epimorphism topology, and they're the same as continous $G$-sets.

Ok, now, can you deduce Grothendieck's theorem from Joyal and Tierney's? I'm not sure, but here's what you can try:

1. Start by proving that if $X$ is connected, so must be $\mathcal G_X$ — this should already bring you closer to a profinite group.

2. Find a categorical interpretation of finite $\pi_1^\text{ét}(X,x)$-sets among all sheaves on the site (I think here "coherent" should be the relevant word, but I'm not sure), and try to see, for a localic groupoid $\mathcal G$, what this categorical interpretation corresponds to in sheaves on $\mathcal G$.

3. Find some properties that $\Sh(X^\text{f,ét})$ "trivially" has that force $\mathcal G_X$ to be a profinite groupoid rather than an arbitrary localic groupoid (by "trivially" here I mean: don't reprove Joyal–Tierney and Grothendieck).

If you can do these 3 things in a suitable way, you should be able to deduce Grothendieck's theorem from Joyal–Tierney's.