This page is an overview of some of the types of "Galois theories" there are. One of the most basic type is the fundamental theorem of covering spaces, which says, roughly, that for each topological space $X$, there is an equivalence of categories $$\mathrm{Cov}(X)\simeq \pi_1(X)\mathbf{Set}.$$ Grothendieck proved an analogue of that statement for schemes $X$: $$\mathrm{EtCov}(X)\simeq \pi_1(X)\mathbf{Set}.$$ (This is again just a very rough formulation and omits some of the assumptions, but you know what I mean.)

I am interested in "topos-theoretic Galois theory". Unfortunately, this section of the nLab page isn't filled out ("(...)"), but I guess that the topos-theoretic formulation of Galois theory states, roughly, that for each topos $\mathcal E$, $$\mathrm{Gal}(\mathcal E)\simeq \pi_1(\mathcal E)\mathbf{Set},\qquad (\ast)$$ where $\mathrm{Gal}(\mathcal E)$ is the full subcategory of $\mathcal E$ consisting of locally constant objects in $\mathcal E$, and $\pi_1(\mathcal E)$ is the fundamental group of $\mathcal E$. (This is suggested by the nLab section "Reformulation of classical Galois theory".)

Question: Is there a reference for $(\ast)$ (and the definition of the fundamental group of a topos which is used here)? Is this in SGA 4?

The linked nLab page fundamental group of a topos refers to (and is mostly copy-pasted from) Porter's paper Abstract Homotopy Theory: The interaction of category theory and homotopy theory, which contains a section called "The fundamental group of a topos", which in turn refers to SGA 1. This is weird, because SGA 1 doesn't discuss topoi, so in particular not the fundamental group of a topos!

The nLab also refers to SGA 4 Exposé IV Exercice 2.7.5 for the definition of the fundamental group and SGA 4 Exposé VIII Proposition 2.1 for, I guess, $(\ast)$ in the special case that $\mathcal E$ is the étale topos of the scheme $X=\mathrm{Spec}(k)$ for some field $k$. (But this is really just a guess - I can't read French. So correct me if I'm wrong.) Is there more of "topos-theoretic Galois theory" in SGA 4 or are these the only two paragraphs about that topic?

Concerning the definition of the fundamental group of a topos, there is a construction in Moerdijk's Classifying Spaces and Classifying Topoi, in which he nevertheless remarks:

The profinite fundamental group is discussed in SGA1.

This suggests there are two version of the fundamental group of a topos: the one he discusses and the "profinite" version. However, as I said, topoi don't occur in SGA 1, so I wonder where I can find the definition of the "profinite" fundamental group, if that's the notion that should be used in $(\ast)$. (The definition used in $(\ast)$ should of course have the property that if $\mathcal E$ is the étale topos of a scheme $X$, then $\pi_1(\mathcal E)$ is isomorphic to the étale fundamental group of $X$.)

  • $\begingroup$ Have you checked 8.4 of Johnstone's (first, 1977) topos theory book? $\endgroup$ Feb 6, 2022 at 18:48
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    $\begingroup$ Grothendieck's Galois theory is limited to finite covering spaces i.e. locally constant sheaves of finite sets. I don't know for which topoi the category of locally constant sheaves of finite sets is a Galois category in Grothendieck's sense. More generally, there is a notion of a (tame) infinite Galois category due to Bhatt and Scholze. I think they show an example of a topos such that locally constant sheaves do not form a tame infinite Galois category. I am sure you can always define the "shape" of a topos as a pro-homotopy type, but I don't know how its $\pi_1$ relates to local systems. $\endgroup$ Feb 6, 2022 at 18:54
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    $\begingroup$ Let me start by a disclaimer, I am no expert in the subject discussed nor am I working in the field, just a curious fellow who was intrigued by those beautiful concepts once and remember a couple of references. Borceux and Janelidze's book "Galois Theories", might be a good place to look. I also remember Olivia Caramello (perhaps with Laurent Lafforgue) did some work in the subject, a simple google search should give you something. There is also an old paper of Barr and Diaconescu "On locally simply connected toposes and their fundamental groups". $\endgroup$
    – Andry
    Feb 6, 2022 at 19:11
  • $\begingroup$ Ah, using Google translator I think Exercice 2.7.5 is exactly the statement $(\ast)$. $\endgroup$ Feb 6, 2022 at 19:37
  • $\begingroup$ I first learned of fundamental groups of topoi from numdam.org/item/?id=CTGDC_1981__22_3_301_0 $\endgroup$ Feb 7, 2022 at 11:27

2 Answers 2


I believe that chapter $5$ of Galois Theories by Borceux and Janelidze contains what you're looking for -- they develop a general 'categorical Galois theorem' invoking toposes.

Chapter 7 gets even more general, presenting a 'non-Galoisian Galois theorem' that holds for any descent morphism in a category -- a morphism $f$ is a descent morphism iff the 'pullback along $f$' functor is monadic.

There is also the classic paper by Joyal and Tierney, An Extension of the Galois theory of Grothendieck, where they prove that each Grothendieck topos is equivalent to the category of equivariant sheaves on a groupoid internal to the category of locales.

This paper by Christopher Townsend might be of interest to your specific question; he re-proves Joyal and Tierney’s result on the representation of Grothendieck toposes as localic groupoids using a simplified case of the aforementioned categorical Galois theorem, then proceeds to actually prove the whole theorem using this trivial case as a key ingredient.

  • $\begingroup$ I don't think this is in Chapter 5 of the book by Borceux et al, because topoi are defined later in chapter 7. There they discuss the Joyal-Tierney result. Does the Joyal-Tierney result imply $(\ast)$? $\endgroup$ Feb 8, 2022 at 16:08
  • $\begingroup$ Also, I don't see how the Townsend paper is related to my specific question. $\endgroup$ Feb 8, 2022 at 16:10
  • $\begingroup$ I have the feeling Joyal-Tierney and $(\ast)$ are somewhat different generalizations of the fundamental theorem of covering spaces. $\endgroup$ Feb 8, 2022 at 16:17
  • $\begingroup$ @user1022117 I'll have to think on wether $(*)$ is a special case of the Joyal-Tierney result or not, but I'm curious about why you're convinced that $(*)$ is the 'correct' generalization? $\endgroup$
    – Alec Rhea
    Feb 8, 2022 at 18:40
  • $\begingroup$ I didn't claim it's the "correct" one. I'm just asking about it. $\endgroup$ Feb 8, 2022 at 18:59

Maybe you would like to see the thesis of O. Leroy, Groupoïde fondamental et théorème de Van Kampen en théorie des topos, available from https://plmbox.math.cnrs.fr/f/7ffa366379144dd4bacc/?dl=1 — the password is : groupoide

(a project of transcription)

or the thesis of V. Zoonekynd, La Tour de Teichmuller-Grothendieck, available from https://tel.archives-ouvertes.fr/tel-00001140/document

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    $\begingroup$ A very clear summary for some of the ideas in Leroy's thesis is in this paper by Zoonekynd: arxiv.org/abs/math/0111071 (Section 1). $\endgroup$ Feb 7, 2022 at 15:29
  • $\begingroup$ Which theorem is it in these documents? $\endgroup$ Feb 8, 2022 at 16:04
  • $\begingroup$ Without a specific reference to a theorem that doesn't at all answer my Question. $\endgroup$ Feb 9, 2022 at 12:16

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