Fundamental groups of topoi

Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given:

If $T$ is a Grothendieck topos arising as category of sheaves on a site $X$, then there is the notion of locally constant, locally finite objects in $T$ (which I presume just means that there is a cover $(U_i)$ in $X$ such that each restriction to $U_i$ is constant and finite). If $C$ is the subcategory of $T$ consisting of all the locally constant, locally finite objects of $T$, and if $F:C\rightarrow FinSets$ is a functor ("fiber functor"), satisfying certain unnamed properties which should imply prorepresentability, then one defines $\pi_1(T,F)=Aut(F)$.

Now, if $X_{et}$ is the small étale site of a connected scheme $X$, then it is well known the category of locally constant, locally finite sheaves on $X$ is equivalent to the category of finite étale coverings of $X$, and with the appropriate notion of fiber functor it surely follows that the étale fundamental group and the fundamental group of the topos on $X_{et}$ coincide.

Similarly, as the nlab entry mentions, if $X$ is a nice topological space, locally finite, locally constant sheaves correspond to finite covering spaces (via the "éspace étalé"), and we should recover the profinite completion of the usual topological fundamental group.

Before I come to my main question: Did I manage to summarize this correctly, or is there something wrong with the above?

My question:

Has the fundamental group of other topoi been studied, and in what context or disguise might we already know them? For example, what is known about the fundamental group of the category of fppf sheaves over a scheme $X$?

-

The profinite fundamental group of $X_{fppf}$ as you define it is again the etale fundamental group of X. More precisely, the functor (of points)

$f : X_{et} \to \mathrm{Sh}_{fppf}(X)$

is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale maps are even etale locally finite constant, let alone fppf locally so). Proof in 3 steps:

1. It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).

2. Both sides are fppf sheaves (stacks) in $X$, by classical fppf descent.

3. Combining 1 and 2, it suffices to show that a sheaf we want to hit is just fppf locally hit, which is obvious since locally it's finite constant.

Note that the same proof also works for $X_{et}$ or anything in between -- once your topology splits finite etale maps it doesn't really matter what it is. So we usually just work with the minimal one, the small etale topology. As Mike Artin said to me apropos of something like this, "Why pack a suitcase when you're just going around the corner?"

-
+1 good answer, and I like the quote! – Harry Gindi Jun 17 '10 at 19:45
As you know, this also works "beyond" fppf with fpqc too. I'm surprised that Harry didn't point this out. :) – Boyarsky Jun 18 '10 at 16:58

In answer to the 'pre-question', yes, pretty much correct. I would just add that the cover $(U_i)$ in X in your second paragraph is a cover of the terminal object in the site $X$, if it has one. If not, then I think it's a bit fiddlier.

Joyal and Tierney showed that every topos is the topos of sheaves on a localic groupoid (a groupoid internal to the category of locales) and this groupoid is basically the fundamental groupoid of that topos. If one assumes successively stronger conditions about the topos, then this groupoid becomes more like the more familiar notions. If the topos has a point (not all do!) then one can talk about the fundamental group (which is in full generality, localic). Then if the topos is locally connected, it gets nicer. Marta Bunge has done a lot of work on this, with various people.

As far as connecting with other notions, I'll let the algebraic geometers answer that.

-
Hi, thanks for the answer. Unfortunately I'm not very fluent in Topoi-language, I will look up what a locale is though. I was aware of the problem that sites don't need to have final objects, that's why I was sketchy :) Maybe one can only talk about $Y$-local properties then, for any $Y$ in the site. Or maybe one embeds the site via yoneda into the topos and uses the a covering of the final object in the topos. – Lars May 4 '10 at 16:26
An object $Y$ in a topos $E$ is called locally constant if there is a cover $U$ of the terminal object and an isomorphism $Y \times U \simeq U \times p^*F$ for some set $F$ and where $p:E \to Set$ is the canonical map. I think for the sort of site you seem to interested in, there is a terminal object (small site of a scheme/space), so that's not too much of a complicating factor, I suppose. – David Roberts May 4 '10 at 23:41
Unfortunately, if I remember correctly, even the cyrstalline site of a non smooth scheme, does not need to have a final object. And it would be interesting to know the fundamental group of the crystalline topos. – Lars May 6 '10 at 6:51
The notion of something being locally true for an object Y in a topos T is: You look at a collection of objects X_i and the slice categories T/X_i (these are topoi again, corresponding to open subsets). You get a collection of geometric morphisms T/X_i-->T (--> given by composing Z->X_i with X_i->1 [1 the terminal object of the topos which always exists], <-- given by associating with Z' the pullback along X_i-->1, i.e. product with X_i+projection). You should see this as a covering if the familiy <-- is jointly conservative, i.e. if two arrows are different in T then they will be so in ... – Peter Arndt May 7 '10 at 16:35
I don't think that just because a topos is the topos of sheaves on a localic groupoid implies that that localic groupoid is the fundamental groupoid of the topos. The localic groupoids that occur as fundamental groupoids of topoi are all prodiscrete, whereas any localic groupoid has a topos of sheaves. – Mike Shulman Feb 5 '11 at 9:12

V. Zoonekynd has defined the etale fundamental group of an algebraic stack using this point of vue.

From what I understand, he associates with a locally connected topos $T$ its topos of sums of locally constant objects $SLC(T)$. This is a "locally galoisian" topos and if $T$ has at least one point in every connected component it is the classifying topos of the fundamental groupoïd of $T$. Inclusion induces a morphism $T \to SLC(T)$ which is universal w/r to morphism to locally galoisian topoi.

-