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 $X$?
Best Answer
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:
It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).
Both sides are fppf sheaves (stacks) in $X$, by classical fppf descent.
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?"