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There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to understand the relationships amongst them.

I have begun to look at their papers and can see some of the interrelations which their prefaces often lay out very well. Still I wonder if there is a relatively comprehensive bird's eye view of the landscape that can orient my reading and understanding as I delve deeper.

If it's too involved to sketch here as an answer, is there a survey article I can look at? Do the monographs of Szamuely or Borceux-Janelidze contain it?

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    $\begingroup$ "fundamental group" is a bit vague IMHO : are you interested in étale fundamental groups, in fundamental group schemes, in motivic fundamental groups, or in whatever ? $\endgroup$
    – Niels
    Commented Sep 24, 2018 at 7:29
  • $\begingroup$ In any case, it looks a bit odd to me not to add Deligne to your list. $\endgroup$
    – Niels
    Commented Sep 24, 2018 at 7:33
  • $\begingroup$ @Niels, I've added Deligne. I hope the survey would include all the different flavors of fundamental groups/groupoids/group-schemes and their inter-connections, as well as motivations for introducing them and their distinct uses. $\endgroup$
    – Galois groupie
    Commented Sep 25, 2018 at 0:16
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    $\begingroup$ This is a very large topic, and I don't know a single source that contains even a significant number of these. Esnault has some nice papers discussing the relationship between the stratified and etale fundamental group; Deligne's P^1\{0,1,\infty} paper discusses the relationship between various pro-unipotent incarnations of these groups. If you have a more specific question in mind, I can try to say a bit more. $\endgroup$
    – Daniel Litt
    Commented Sep 25, 2018 at 4:30

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